Download - 2: Three Different Fourier Transforms
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2: Three Different FourierTransforms
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 1 / 14
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Fourier Transforms
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 2 / 14
Three different Fourier Transforms:
![Page 3: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/3.jpg)
Fourier Transforms
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 2 / 14
Three different Fourier Transforms:
• CTFT (Continuous-Time Fourier Transform): x(t) → X(jΩ)
![Page 4: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/4.jpg)
Fourier Transforms
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 2 / 14
Three different Fourier Transforms:
• CTFT (Continuous-Time Fourier Transform): x(t) → X(jΩ)• DTFT (Discrete-Time Fourier Transform): x[n] → X(ejω)
![Page 5: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/5.jpg)
Fourier Transforms
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 2 / 14
Three different Fourier Transforms:
• CTFT (Continuous-Time Fourier Transform): x(t) → X(jΩ)• DTFT (Discrete-Time Fourier Transform): x[n] → X(ejω)• DFT a.k.a. FFT (Discrete Fourier Transform): x[n] → X[k]
![Page 6: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/6.jpg)
Fourier Transforms
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 2 / 14
Three different Fourier Transforms:
• CTFT (Continuous-Time Fourier Transform): x(t) → X(jΩ)• DTFT (Discrete-Time Fourier Transform): x[n] → X(ejω)• DFT a.k.a. FFT (Discrete Fourier Transform): x[n] → X[k]
Forward Transform Inverse Transform
CTFT X(jΩ) =∫∞−∞ x(t)e−jΩtdt x(t) = 1
2π
∫∞−∞ X(jΩ)ejΩtdΩ
![Page 7: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/7.jpg)
Fourier Transforms
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 2 / 14
Three different Fourier Transforms:
• CTFT (Continuous-Time Fourier Transform): x(t) → X(jΩ)• DTFT (Discrete-Time Fourier Transform): x[n] → X(ejω)• DFT a.k.a. FFT (Discrete Fourier Transform): x[n] → X[k]
Forward Transform Inverse Transform
CTFT X(jΩ) =∫∞−∞ x(t)e−jΩtdt x(t) = 1
2π
∫∞−∞ X(jΩ)ejΩtdΩ
DTFT X(ejω) =∑∞
−∞ x[n]e−jωn x[n] = 12π
∫ π
−πX(ejω)ejωndω
![Page 8: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/8.jpg)
Fourier Transforms
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 2 / 14
Three different Fourier Transforms:
• CTFT (Continuous-Time Fourier Transform): x(t) → X(jΩ)• DTFT (Discrete-Time Fourier Transform): x[n] → X(ejω)• DFT a.k.a. FFT (Discrete Fourier Transform): x[n] → X[k]
Forward Transform Inverse Transform
CTFT X(jΩ) =∫∞−∞ x(t)e−jΩtdt x(t) = 1
2π
∫∞−∞ X(jΩ)ejΩtdΩ
DTFT X(ejω) =∑∞
−∞ x[n]e−jωn x[n] = 12π
∫ π
−πX(ejω)ejωndω
We use Ω for “real” and ω = ΩT for “normalized” angular frequency.Nyquist frequency is at ΩNyq = 2π fs
2 = πT
and ωNyq = π.
![Page 9: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/9.jpg)
Fourier Transforms
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 2 / 14
Three different Fourier Transforms:
• CTFT (Continuous-Time Fourier Transform): x(t) → X(jΩ)• DTFT (Discrete-Time Fourier Transform): x[n] → X(ejω)• DFT a.k.a. FFT (Discrete Fourier Transform): x[n] → X[k]
Forward Transform Inverse Transform
CTFT X(jΩ) =∫∞−∞ x(t)e−jΩtdt x(t) = 1
2π
∫∞−∞ X(jΩ)ejΩtdΩ
DTFT X(ejω) =∑∞
−∞ x[n]e−jωn x[n] = 12π
∫ π
−πX(ejω)ejωndω
DFT X[k] =∑N−1
0 x[n]e−j2π kn
N x[n] = 1N
∑N−10 X[k]ej2π
kn
N
We use Ω for “real” and ω = ΩT for “normalized” angular frequency.Nyquist frequency is at ΩNyq = 2π fs
2 = πT
and ωNyq = π.
![Page 10: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/10.jpg)
Fourier Transforms
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 2 / 14
Three different Fourier Transforms:
• CTFT (Continuous-Time Fourier Transform): x(t) → X(jΩ)• DTFT (Discrete-Time Fourier Transform): x[n] → X(ejω)• DFT a.k.a. FFT (Discrete Fourier Transform): x[n] → X[k]
Forward Transform Inverse Transform
CTFT X(jΩ) =∫∞−∞ x(t)e−jΩtdt x(t) = 1
2π
∫∞−∞ X(jΩ)ejΩtdΩ
DTFT X(ejω) =∑∞
−∞ x[n]e−jωn x[n] = 12π
∫ π
−πX(ejω)ejωndω
DFT X[k] =∑N−1
0 x[n]e−j2π kn
N x[n] = 1N
∑N−10 X[k]ej2π
kn
N
We use Ω for “real” and ω = ΩT for “normalized” angular frequency.Nyquist frequency is at ΩNyq = 2π fs
2 = πT
and ωNyq = π.
For “power signals” (energy ∝ duration), CTFT & DTFT are unbounded.Fix this by normalizing:
X(jΩ) = limA→∞12A
∫ A
−Ax(t)e−jΩtdt
X(ejω) = limA→∞1
2A+1
∑A−A x[n]e−jωn
![Page 11: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/11.jpg)
Convergence of DTFT
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 3 / 14
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn does not converge for all x[n].
Consider the finite sum: XK(ejω) =∑K
−K x[n]e−jωn
![Page 12: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/12.jpg)
Convergence of DTFT
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 3 / 14
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn does not converge for all x[n].
Consider the finite sum: XK(ejω) =∑K
−K x[n]e−jωn
Strong Convergence:x[n] absolutely summable ⇒X(ejω) converges uniformly
![Page 13: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/13.jpg)
Convergence of DTFT
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 3 / 14
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn does not converge for all x[n].
Consider the finite sum: XK(ejω) =∑K
−K x[n]e−jωn
Strong Convergence:x[n] absolutely summable ⇒X(ejω) converges uniformly∑∞
−∞ |x[n]| < ∞⇒ supω∣
∣X(ejω)−XK(ejω)∣
∣ −−−−→K→∞
0
![Page 14: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/14.jpg)
Convergence of DTFT
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 3 / 14
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn does not converge for all x[n].
Consider the finite sum: XK(ejω) =∑K
−K x[n]e−jωn
Strong Convergence:x[n] absolutely summable ⇒X(ejω) converges uniformly∑∞
−∞ |x[n]| < ∞⇒ supω∣
∣X(ejω)−XK(ejω)∣
∣ −−−−→K→∞
0
Weaker convergence:x[n] finite energy ⇒X(ejω) converges in mean square
![Page 15: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/15.jpg)
Convergence of DTFT
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 3 / 14
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn does not converge for all x[n].
Consider the finite sum: XK(ejω) =∑K
−K x[n]e−jωn
Strong Convergence:x[n] absolutely summable ⇒X(ejω) converges uniformly∑∞
−∞ |x[n]| < ∞⇒ supω∣
∣X(ejω)−XK(ejω)∣
∣ −−−−→K→∞
0
Weaker convergence:x[n] finite energy ⇒X(ejω) converges in mean square∑∞
−∞ |x[n]|2< ∞⇒ 1
2π
∫ π
−π
∣
∣X(ejω)−XK(ejω)∣
∣
2dω −−−−→
K→∞0
![Page 16: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/16.jpg)
Convergence of DTFT
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 3 / 14
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn does not converge for all x[n].
Consider the finite sum: XK(ejω) =∑K
−K x[n]e−jωn
Strong Convergence:x[n] absolutely summable ⇒X(ejω) converges uniformly∑∞
−∞ |x[n]| < ∞⇒ supω∣
∣X(ejω)−XK(ejω)∣
∣ −−−−→K→∞
0
Weaker convergence:x[n] finite energy ⇒X(ejω) converges in mean square∑∞
−∞ |x[n]|2< ∞⇒ 1
2π
∫ π
−π
∣
∣X(ejω)−XK(ejω)∣
∣
2dω −−−−→
K→∞0
Example: x[n] = sin 0.5πnπn
0 0.1 0.2 0.3 0.4 0.5-0.2
0
0.2
0.4
0.6
0.8
1
ω/2π (rad/sample)
K j ω
K=5
0 0.1 0.2 0.3 0.4 0.5-0.2
0
0.2
0.4
0.6
0.8
1
ω/2π (rad/sample)
K j ω
K=20
0 0.1 0.2 0.3 0.4 0.5-0.2
0
0.2
0.4
0.6
0.8
1
ω/2π (rad/sample)
K j ω
K=50
![Page 17: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/17.jpg)
Convergence of DTFT
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 3 / 14
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn does not converge for all x[n].
Consider the finite sum: XK(ejω) =∑K
−K x[n]e−jωn
Strong Convergence:x[n] absolutely summable ⇒X(ejω) converges uniformly∑∞
−∞ |x[n]| < ∞⇒ supω∣
∣X(ejω)−XK(ejω)∣
∣ −−−−→K→∞
0
Weaker convergence:x[n] finite energy ⇒X(ejω) converges in mean square∑∞
−∞ |x[n]|2< ∞⇒ 1
2π
∫ π
−π
∣
∣X(ejω)−XK(ejω)∣
∣
2dω −−−−→
K→∞0
Example: x[n] = sin 0.5πnπn
0 0.1 0.2 0.3 0.4 0.5-0.2
0
0.2
0.4
0.6
0.8
1
ω/2π (rad/sample)
K j ω
K=5
0 0.1 0.2 0.3 0.4 0.5-0.2
0
0.2
0.4
0.6
0.8
1
ω/2π (rad/sample)
K j ω
K=20
0 0.1 0.2 0.3 0.4 0.5-0.2
0
0.2
0.4
0.6
0.8
1
ω/2π (rad/sample)
K j ω
K=50
Gibbs phenomenon:Converges at each ω as K → ∞ but peak error does not get smaller.
![Page 18: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/18.jpg)
DTFT Properties
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 4 / 14
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn
![Page 19: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/19.jpg)
DTFT Properties
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 4 / 14
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn
• DTFT is periodic in ω: X(ej(ω+2mπ)) = X(ejω) for integer m.
![Page 20: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/20.jpg)
DTFT Properties
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 4 / 14
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn
• DTFT is periodic in ω: X(ej(ω+2mπ)) = X(ejω) for integer m.
• DTFT is the z-Transform evaluated at the point ejω :X(z) =
∑∞−∞ x[n]z−n
![Page 21: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/21.jpg)
DTFT Properties
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 4 / 14
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn
• DTFT is periodic in ω: X(ej(ω+2mπ)) = X(ejω) for integer m.
• DTFT is the z-Transform evaluated at the point ejω :X(z) =
∑∞−∞ x[n]z−n
DTFT converges iff the ROC includes |z| = 1.
![Page 22: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/22.jpg)
DTFT Properties
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 4 / 14
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn
• DTFT is periodic in ω: X(ej(ω+2mπ)) = X(ejω) for integer m.
• DTFT is the z-Transform evaluated at the point ejω :X(z) =
∑∞−∞ x[n]z−n
DTFT converges iff the ROC includes |z| = 1.
• DTFT is the same as the CTFT of a signal comprising impulses atthe sample times (Dirac δ functions) of appropriate heights:
xδ(t) =∑
x[n]δ(t− nT )
![Page 23: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/23.jpg)
DTFT Properties
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 4 / 14
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn
• DTFT is periodic in ω: X(ej(ω+2mπ)) = X(ejω) for integer m.
• DTFT is the z-Transform evaluated at the point ejω :X(z) =
∑∞−∞ x[n]z−n
DTFT converges iff the ROC includes |z| = 1.
• DTFT is the same as the CTFT of a signal comprising impulses atthe sample times (Dirac δ functions) of appropriate heights:
xδ(t) =∑
x[n]δ(t− nT )Equivalent to multiplying a continuous x(t) by an impulse train.
![Page 24: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/24.jpg)
DTFT Properties
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 4 / 14
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn
• DTFT is periodic in ω: X(ej(ω+2mπ)) = X(ejω) for integer m.
• DTFT is the z-Transform evaluated at the point ejω :X(z) =
∑∞−∞ x[n]z−n
DTFT converges iff the ROC includes |z| = 1.
• DTFT is the same as the CTFT of a signal comprising impulses atthe sample times (Dirac δ functions) of appropriate heights:
xδ(t) =∑
x[n]δ(t− nT )= x(t)×∑∞
−∞ δ(t− nT )Equivalent to multiplying a continuous x(t) by an impulse train.
![Page 25: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/25.jpg)
DTFT Properties
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 4 / 14
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn
• DTFT is periodic in ω: X(ej(ω+2mπ)) = X(ejω) for integer m.
• DTFT is the z-Transform evaluated at the point ejω :X(z) =
∑∞−∞ x[n]z−n
DTFT converges iff the ROC includes |z| = 1.
• DTFT is the same as the CTFT of a signal comprising impulses atthe sample times (Dirac δ functions) of appropriate heights:
xδ(t) =∑
x[n]δ(t− nT )= x(t)×∑∞
−∞ δ(t− nT )Equivalent to multiplying a continuous x(t) by an impulse train.
Proof: X(ejω) =∑∞
−∞ x[n]e−jωn
![Page 26: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/26.jpg)
DTFT Properties
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 4 / 14
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn
• DTFT is periodic in ω: X(ej(ω+2mπ)) = X(ejω) for integer m.
• DTFT is the z-Transform evaluated at the point ejω :X(z) =
∑∞−∞ x[n]z−n
DTFT converges iff the ROC includes |z| = 1.
• DTFT is the same as the CTFT of a signal comprising impulses atthe sample times (Dirac δ functions) of appropriate heights:
xδ(t) =∑
x[n]δ(t− nT )= x(t)×∑∞
−∞ δ(t− nT )Equivalent to multiplying a continuous x(t) by an impulse train.
Proof: X(ejω) =∑∞
−∞ x[n]e−jωn
∑∞n=−∞ x[n]
∫∞−∞ δ(t− nT )e−jω t
T dt
![Page 27: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/27.jpg)
DTFT Properties
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 4 / 14
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn
• DTFT is periodic in ω: X(ej(ω+2mπ)) = X(ejω) for integer m.
• DTFT is the z-Transform evaluated at the point ejω :X(z) =
∑∞−∞ x[n]z−n
DTFT converges iff the ROC includes |z| = 1.
• DTFT is the same as the CTFT of a signal comprising impulses atthe sample times (Dirac δ functions) of appropriate heights:
xδ(t) =∑
x[n]δ(t− nT )= x(t)×∑∞
−∞ δ(t− nT )Equivalent to multiplying a continuous x(t) by an impulse train.
Proof: X(ejω) =∑∞
−∞ x[n]e−jωn
∑∞n=−∞ x[n]
∫∞−∞ δ(t− nT )e−jω t
T dt
(i)=
∫∞−∞
∑∞n=−∞ x[n]δ(t− nT )e−jω t
T dt
(i) OK if∑∞
−∞ |x[n]| < ∞.
![Page 28: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/28.jpg)
DTFT Properties
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 4 / 14
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn
• DTFT is periodic in ω: X(ej(ω+2mπ)) = X(ejω) for integer m.
• DTFT is the z-Transform evaluated at the point ejω :X(z) =
∑∞−∞ x[n]z−n
DTFT converges iff the ROC includes |z| = 1.
• DTFT is the same as the CTFT of a signal comprising impulses atthe sample times (Dirac δ functions) of appropriate heights:
xδ(t) =∑
x[n]δ(t− nT )= x(t)×∑∞
−∞ δ(t− nT )Equivalent to multiplying a continuous x(t) by an impulse train.
Proof: X(ejω) =∑∞
−∞ x[n]e−jωn
∑∞n=−∞ x[n]
∫∞−∞ δ(t− nT )e−jω t
T dt
(i)=
∫∞−∞
∑∞n=−∞ x[n]δ(t− nT )e−jω t
T dt
(ii)=
∫∞−∞ xδ(t)e
−jΩtdt
(i) OK if∑∞
−∞ |x[n]| < ∞. (ii) use ω = ΩT .
![Page 29: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/29.jpg)
DFT Properties
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 5 / 14
DFT: X[k] =∑N−1
0 x[n]e−j2π kn
N
![Page 30: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/30.jpg)
DFT Properties
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 5 / 14
DFT: X[k] =∑N−1
0 x[n]e−j2π kn
N
Case 1: x[n] = 0 for n /∈ [0, N − 1]
DFT is the same as DTFT at ωk = 2πNk.
![Page 31: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/31.jpg)
DFT Properties
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 5 / 14
DFT: X[k] =∑N−1
0 x[n]e−j2π kn
N
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn
Case 1: x[n] = 0 for n /∈ [0, N − 1]
DFT is the same as DTFT at ωk = 2πNk.
![Page 32: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/32.jpg)
DFT Properties
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 5 / 14
DFT: X[k] =∑N−1
0 x[n]e−j2π kn
N
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn
Case 1: x[n] = 0 for n /∈ [0, N − 1]
DFT is the same as DTFT at ωk = 2πNk.
The ωk are uniformly spaced from ω = 0 to ω = 2πN−1N
.
![Page 33: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/33.jpg)
DFT Properties
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 5 / 14
DFT: X[k] =∑N−1
0 x[n]e−j2π kn
N
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn
Case 1: x[n] = 0 for n /∈ [0, N − 1]
DFT is the same as DTFT at ωk = 2πNk.
The ωk are uniformly spaced from ω = 0 to ω = 2πN−1N
.DFT is the z-Transform evaluated at N equally spaced pointsaround the unit circle beginning at z = 1.
![Page 34: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/34.jpg)
DFT Properties
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 5 / 14
DFT: X[k] =∑N−1
0 x[n]e−j2π kn
N
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn
Case 1: x[n] = 0 for n /∈ [0, N − 1]
DFT is the same as DTFT at ωk = 2πNk.
The ωk are uniformly spaced from ω = 0 to ω = 2πN−1N
.DFT is the z-Transform evaluated at N equally spaced pointsaround the unit circle beginning at z = 1.
Case 2: x[n] is periodic with period N
DFT equals the normalized DTFT
![Page 35: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/35.jpg)
DFT Properties
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 5 / 14
DFT: X[k] =∑N−1
0 x[n]e−j2π kn
N
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn
Case 1: x[n] = 0 for n /∈ [0, N − 1]
DFT is the same as DTFT at ωk = 2πNk.
The ωk are uniformly spaced from ω = 0 to ω = 2πN−1N
.DFT is the z-Transform evaluated at N equally spaced pointsaround the unit circle beginning at z = 1.
Case 2: x[n] is periodic with period N
DFT equals the normalized DTFT
X[k] = limK→∞N
2K+1 ×XK(ejωk)
![Page 36: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/36.jpg)
DFT Properties
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 5 / 14
DFT: X[k] =∑N−1
0 x[n]e−j2π kn
N
DTFT: X(ejω) =∑∞
−∞ x[n]e−jωn
Case 1: x[n] = 0 for n /∈ [0, N − 1]
DFT is the same as DTFT at ωk = 2πNk.
The ωk are uniformly spaced from ω = 0 to ω = 2πN−1N
.DFT is the z-Transform evaluated at N equally spaced pointsaround the unit circle beginning at z = 1.
Case 2: x[n] is periodic with period N
DFT equals the normalized DTFT
X[k] = limK→∞N
2K+1 ×XK(ejωk)
where XK(ejω) =∑K
−K x[n]e−jωn
![Page 37: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/37.jpg)
Symmetries
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 6 / 14
If x[n] has a special property then X(ejω)and X[k] will havecorresponding properties as shown in the table (and vice versa):
One domain Other domain
Discrete PeriodicSymmetric Symmetric
Antisymmetric AntisymmetricReal Conjugate Symmetric
Imaginary Conjugate AntisymmetricReal + Symmetric Real + Symmetric
Real + Antisymmetric Imaginary + Antisymmetric
![Page 38: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/38.jpg)
Symmetries
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 6 / 14
If x[n] has a special property then X(ejω)and X[k] will havecorresponding properties as shown in the table (and vice versa):
One domain Other domain
Discrete PeriodicSymmetric Symmetric
Antisymmetric AntisymmetricReal Conjugate Symmetric
Imaginary Conjugate AntisymmetricReal + Symmetric Real + Symmetric
Real + Antisymmetric Imaginary + Antisymmetric
Symmetric: x[n] = x[−n]
![Page 39: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/39.jpg)
Symmetries
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 6 / 14
If x[n] has a special property then X(ejω)and X[k] will havecorresponding properties as shown in the table (and vice versa):
One domain Other domain
Discrete PeriodicSymmetric Symmetric
Antisymmetric AntisymmetricReal Conjugate Symmetric
Imaginary Conjugate AntisymmetricReal + Symmetric Real + Symmetric
Real + Antisymmetric Imaginary + Antisymmetric
Symmetric: x[n] = x[−n]X(ejω) = X(e−jω)
![Page 40: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/40.jpg)
Symmetries
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 6 / 14
If x[n] has a special property then X(ejω)and X[k] will havecorresponding properties as shown in the table (and vice versa):
One domain Other domain
Discrete PeriodicSymmetric Symmetric
Antisymmetric AntisymmetricReal Conjugate Symmetric
Imaginary Conjugate AntisymmetricReal + Symmetric Real + Symmetric
Real + Antisymmetric Imaginary + Antisymmetric
Symmetric: x[n] = x[−n]X(ejω) = X(e−jω)X[k] = X[(−k)mod N ] = X[N − k] for k > 0
![Page 41: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/41.jpg)
Symmetries
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 6 / 14
If x[n] has a special property then X(ejω)and X[k] will havecorresponding properties as shown in the table (and vice versa):
One domain Other domain
Discrete PeriodicSymmetric Symmetric
Antisymmetric AntisymmetricReal Conjugate Symmetric
Imaginary Conjugate AntisymmetricReal + Symmetric Real + Symmetric
Real + Antisymmetric Imaginary + Antisymmetric
Symmetric: x[n] = x[−n]X(ejω) = X(e−jω)X[k] = X[(−k)mod N ] = X[N − k] for k > 0
Conjugate Symmetric: x[n] = x∗[−n]
![Page 42: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/42.jpg)
Symmetries
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 6 / 14
If x[n] has a special property then X(ejω)and X[k] will havecorresponding properties as shown in the table (and vice versa):
One domain Other domain
Discrete PeriodicSymmetric Symmetric
Antisymmetric AntisymmetricReal Conjugate Symmetric
Imaginary Conjugate AntisymmetricReal + Symmetric Real + Symmetric
Real + Antisymmetric Imaginary + Antisymmetric
Symmetric: x[n] = x[−n]X(ejω) = X(e−jω)X[k] = X[(−k)mod N ] = X[N − k] for k > 0
Conjugate Symmetric: x[n] = x∗[−n]Conjugate Antisymmetric: x[n] = −x∗[−n]
![Page 43: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/43.jpg)
Parseval’s Theorem
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 7 / 14
Fourier transforms preserve “energy”
![Page 44: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/44.jpg)
Parseval’s Theorem
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 7 / 14
Fourier transforms preserve “energy”
CTFT∫
|x(t)|2 dt = 12π
∫
|X(jΩ)|2 dΩ
![Page 45: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/45.jpg)
Parseval’s Theorem
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 7 / 14
Fourier transforms preserve “energy”
CTFT∫
|x(t)|2 dt = 12π
∫
|X(jΩ)|2 dΩ
DTFT∑∞
−∞ |x[n]|2 = 12π
∫ π
−π
∣
∣X(ejω)∣
∣
2dω
![Page 46: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/46.jpg)
Parseval’s Theorem
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 7 / 14
Fourier transforms preserve “energy”
CTFT∫
|x(t)|2 dt = 12π
∫
|X(jΩ)|2 dΩ
DTFT∑∞
−∞ |x[n]|2 = 12π
∫ π
−π
∣
∣X(ejω)∣
∣
2dω
DFT∑N−1
0 |x[n]|2 = 1N
∑N−10 |X[k]|2
![Page 47: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/47.jpg)
Parseval’s Theorem
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 7 / 14
Fourier transforms preserve “energy”
CTFT∫
|x(t)|2 dt = 12π
∫
|X(jΩ)|2 dΩ
DTFT∑∞
−∞ |x[n]|2 = 12π
∫ π
−π
∣
∣X(ejω)∣
∣
2dω
DFT∑N−1
0 |x[n]|2 = 1N
∑N−10 |X[k]|2
More generally, they actually preserve complex inner products:
∑N−10 x[n]y∗[n] = 1
N
∑N−10 X[k]Y ∗[k]
![Page 48: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/48.jpg)
Parseval’s Theorem
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 7 / 14
Fourier transforms preserve “energy”
CTFT∫
|x(t)|2 dt = 12π
∫
|X(jΩ)|2 dΩ
DTFT∑∞
−∞ |x[n]|2 = 12π
∫ π
−π
∣
∣X(ejω)∣
∣
2dω
DFT∑N−1
0 |x[n]|2 = 1N
∑N−10 |X[k]|2
More generally, they actually preserve complex inner products:
∑N−10 x[n]y∗[n] = 1
N
∑N−10 X[k]Y ∗[k]
Unitary matrix viewpoint for DFT:
If we regard x and X as vectors, then X = Fx where F isa symmetric matrix defined by fk+1,n+1 = e−j2π kn
N .
![Page 49: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/49.jpg)
Parseval’s Theorem
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 7 / 14
Fourier transforms preserve “energy”
CTFT∫
|x(t)|2 dt = 12π
∫
|X(jΩ)|2 dΩ
DTFT∑∞
−∞ |x[n]|2 = 12π
∫ π
−π
∣
∣X(ejω)∣
∣
2dω
DFT∑N−1
0 |x[n]|2 = 1N
∑N−10 |X[k]|2
More generally, they actually preserve complex inner products:
∑N−10 x[n]y∗[n] = 1
N
∑N−10 X[k]Y ∗[k]
Unitary matrix viewpoint for DFT:
If we regard x and X as vectors, then X = Fx where F isa symmetric matrix defined by fk+1,n+1 = e−j2π kn
N .
The inverse DFT matrix is F−1 = 1NF
H
equivalently, G = 1√NF is a unitary matrix with G
HG = I.
![Page 50: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/50.jpg)
Convolution
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 8 / 14
DTFT: Convolution → Product
![Page 51: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/51.jpg)
Convolution
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 8 / 14
DTFT: Convolution → Productx[n] = g[n] ∗ h[n]=
∑∞k=−∞ g[k]h[n− k]
![Page 52: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/52.jpg)
Convolution
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 8 / 14
DTFT: Convolution → Productx[n] = g[n] ∗ h[n]=
∑∞k=−∞ g[k]h[n− k]
⇒ X(ejω) = G(ejω)H(ejω)
![Page 53: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/53.jpg)
Convolution
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 8 / 14
DTFT: Convolution → Productx[n] = g[n] ∗ h[n]=
∑∞k=−∞ g[k]h[n− k]
⇒ X(ejω) = G(ejω)H(ejω)
DFT: Circular convolution→ Productx[n] = g[n]⊛N h[n]=
∑N−1k=0 g[k]h[(n− k)modN ]
![Page 54: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/54.jpg)
Convolution
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 8 / 14
DTFT: Convolution → Productx[n] = g[n] ∗ h[n]=
∑∞k=−∞ g[k]h[n− k]
⇒ X(ejω) = G(ejω)H(ejω)
DFT: Circular convolution→ Productx[n] = g[n]⊛N h[n]=
∑N−1k=0 g[k]h[(n− k)modN ]
⇒ X[k] = G[k]H[k]
![Page 55: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/55.jpg)
Convolution
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 8 / 14
DTFT: Convolution → Productx[n] = g[n] ∗ h[n]=
∑∞k=−∞ g[k]h[n− k]
⇒ X(ejω) = G(ejω)H(ejω)
DFT: Circular convolution→ Productx[n] = g[n]⊛N h[n]=
∑N−1k=0 g[k]h[(n− k)modN ]
⇒ X[k] = G[k]H[k]
g[n] : h[n] :
![Page 56: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/56.jpg)
Convolution
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 8 / 14
DTFT: Convolution → Productx[n] = g[n] ∗ h[n]=
∑∞k=−∞ g[k]h[n− k]
⇒ X(ejω) = G(ejω)H(ejω)
DFT: Circular convolution→ Productx[n] = g[n]⊛N h[n]=
∑N−1k=0 g[k]h[(n− k)modN ]
⇒ X[k] = G[k]H[k]
g[n] : h[n] : g[n] ∗ h[n] :
![Page 57: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/57.jpg)
Convolution
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 8 / 14
DTFT: Convolution → Productx[n] = g[n] ∗ h[n]=
∑∞k=−∞ g[k]h[n− k]
⇒ X(ejω) = G(ejω)H(ejω)
DFT: Circular convolution→ Productx[n] = g[n]⊛N h[n]=
∑N−1k=0 g[k]h[(n− k)modN ]
⇒ X[k] = G[k]H[k]
g[n] : h[n] : g[n] ∗ h[n] : g[n]⊛3 h[n]
![Page 58: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/58.jpg)
Convolution
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 8 / 14
DTFT: Convolution → Productx[n] = g[n] ∗ h[n]=
∑∞k=−∞ g[k]h[n− k]
⇒ X(ejω) = G(ejω)H(ejω)
DFT: Circular convolution→ Productx[n] = g[n]⊛N h[n]=
∑N−1k=0 g[k]h[(n− k)modN ]
⇒ X[k] = G[k]H[k]
DTFT: Product→ Circular Convolution ÷2πy[n] = g[n]h[n]
g[n] : h[n] : g[n] ∗ h[n] : g[n]⊛3 h[n]
![Page 59: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/59.jpg)
Convolution
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 8 / 14
DTFT: Convolution → Productx[n] = g[n] ∗ h[n]=
∑∞k=−∞ g[k]h[n− k]
⇒ X(ejω) = G(ejω)H(ejω)
DFT: Circular convolution→ Productx[n] = g[n]⊛N h[n]=
∑N−1k=0 g[k]h[(n− k)modN ]
⇒ X[k] = G[k]H[k]
DTFT: Product→ Circular Convolution ÷2πy[n] = g[n]h[n]⇒ Y (ejω) = 1
2πG(ejω)⊛π H(ejω) =12π
∫ π
−πG(ejθ)H(ej(ω−θ))dθ
g[n] : h[n] : g[n] ∗ h[n] : g[n]⊛3 h[n]
![Page 60: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/60.jpg)
Convolution
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 8 / 14
DTFT: Convolution → Productx[n] = g[n] ∗ h[n]=
∑∞k=−∞ g[k]h[n− k]
⇒ X(ejω) = G(ejω)H(ejω)
DFT: Circular convolution→ Productx[n] = g[n]⊛N h[n]=
∑N−1k=0 g[k]h[(n− k)modN ]
⇒ X[k] = G[k]H[k]
DTFT: Product→ Circular Convolution ÷2πy[n] = g[n]h[n]⇒ Y (ejω) = 1
2πG(ejω)⊛π H(ejω) =12π
∫ π
−πG(ejθ)H(ej(ω−θ))dθ
DFT: Product→ Circular Convolution ÷Ny[n] = g[n]h[n]
g[n] : h[n] : g[n] ∗ h[n] : g[n]⊛3 h[n]
![Page 61: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/61.jpg)
Convolution
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 8 / 14
DTFT: Convolution → Productx[n] = g[n] ∗ h[n]=
∑∞k=−∞ g[k]h[n− k]
⇒ X(ejω) = G(ejω)H(ejω)
DFT: Circular convolution→ Productx[n] = g[n]⊛N h[n]=
∑N−1k=0 g[k]h[(n− k)modN ]
⇒ X[k] = G[k]H[k]
DTFT: Product→ Circular Convolution ÷2πy[n] = g[n]h[n]⇒ Y (ejω) = 1
2πG(ejω)⊛π H(ejω) =12π
∫ π
−πG(ejθ)H(ej(ω−θ))dθ
DFT: Product→ Circular Convolution ÷Ny[n] = g[n]h[n]
⇒ Y [k] = 1NG[k]⊛N H[k]
g[n] : h[n] : g[n] ∗ h[n] : g[n]⊛3 h[n]
![Page 62: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/62.jpg)
Sampling Process
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 9 / 14
Time Time Frequency
AnalogCTFT−→
![Page 63: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/63.jpg)
Sampling Process
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 9 / 14
Time Time Frequency
AnalogCTFT−→
Low PassFilter
* →CTFT−→
![Page 64: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/64.jpg)
Sampling Process
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 9 / 14
Time Time Frequency
AnalogCTFT−→
Low PassFilter
* →CTFT−→
Sample × →DTFT−→
![Page 65: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/65.jpg)
Sampling Process
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 9 / 14
Time Time Frequency
AnalogCTFT−→
Low PassFilter
* →CTFT−→
Sample × →DTFT−→
Window × →DTFT−→
![Page 66: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/66.jpg)
Sampling Process
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 9 / 14
Time Time Frequency
AnalogCTFT−→
Low PassFilter
* →CTFT−→
Sample × →DTFT−→
Window × →DTFT−→
DFTDFT−→
![Page 67: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/67.jpg)
Zero-Padding
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 10 / 14
Zero padding means added extra zeros onto the end of x[n] beforeperforming the DFT.
Time x[n]
WindowedSignal
![Page 68: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/68.jpg)
Zero-Padding
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 10 / 14
Zero padding means added extra zeros onto the end of x[n] beforeperforming the DFT.
Time x[n]
WindowedSignal
With zero-padding
![Page 69: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/69.jpg)
Zero-Padding
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 10 / 14
Zero padding means added extra zeros onto the end of x[n] beforeperforming the DFT.
Time x[n] Frequency |X[k]|
WindowedSignal
With zero-padding
![Page 70: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/70.jpg)
Zero-Padding
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 10 / 14
Zero padding means added extra zeros onto the end of x[n] beforeperforming the DFT.
Time x[n] Frequency |X[k]|
WindowedSignal
With zero-padding
![Page 71: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/71.jpg)
Zero-Padding
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 10 / 14
Zero padding means added extra zeros onto the end of x[n] beforeperforming the DFT.
Time x[n] Frequency |X[k]|
WindowedSignal
With zero-padding
• Zero-padding causes the DFT to evaluate the DTFT at more valuesof ωk. Denser frequency samples.
![Page 72: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/72.jpg)
Zero-Padding
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 10 / 14
Zero padding means added extra zeros onto the end of x[n] beforeperforming the DFT.
Time x[n] Frequency |X[k]|
WindowedSignal
With zero-padding
• Zero-padding causes the DFT to evaluate the DTFT at more valuesof ωk. Denser frequency samples.
• Width of the peaks remains constant: determined by the length andshape of the window.
![Page 73: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/73.jpg)
Zero-Padding
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 10 / 14
Zero padding means added extra zeros onto the end of x[n] beforeperforming the DFT.
Time x[n] Frequency |X[k]|
WindowedSignal
With zero-padding
• Zero-padding causes the DFT to evaluate the DTFT at more valuesof ωk. Denser frequency samples.
• Width of the peaks remains constant: determined by the length andshape of the window.
• Smoother graph but increased frequency resolution is an illusion.
![Page 74: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/74.jpg)
Phase Unwrapping
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 11 / 14
Phase of a DTFT is only defined to within an integer multiple of 2π.
x[n] |X[k]|
![Page 75: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/75.jpg)
Phase Unwrapping
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 11 / 14
Phase of a DTFT is only defined to within an integer multiple of 2π.
x[n] |X[k]|
∠X[k]
![Page 76: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/76.jpg)
Phase Unwrapping
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 11 / 14
Phase of a DTFT is only defined to within an integer multiple of 2π.
x[n] |X[k]|
∠X[k]
Phase unwrapping adds multiples of 2π onto each ∠X[k] to make thephase as continuous as possible.
![Page 77: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/77.jpg)
Phase Unwrapping
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 11 / 14
Phase of a DTFT is only defined to within an integer multiple of 2π.
x[n] |X[k]|
∠X[k] ∠X[k] unwrapped
Phase unwrapping adds multiples of 2π onto each ∠X[k] to make thephase as continuous as possible.
![Page 78: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/78.jpg)
Uncertainty principle
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 12 / 14
CTFT uncertainty principle:( ∫
t2|x(t)|2dt∫|x(t)|2dt
)1
2
( ∫ω2|X(jω)|2dω∫|X(jω)|2dω
)1
2
≥ 12
![Page 79: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/79.jpg)
Uncertainty principle
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 12 / 14
CTFT uncertainty principle:( ∫
t2|x(t)|2dt∫|x(t)|2dt
)1
2
( ∫ω2|X(jω)|2dω∫|X(jω)|2dω
)1
2
≥ 12
The first term measures the “width” of x(t) around t = 0.
![Page 80: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/80.jpg)
Uncertainty principle
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 12 / 14
CTFT uncertainty principle:( ∫
t2|x(t)|2dt∫|x(t)|2dt
)1
2
( ∫ω2|X(jω)|2dω∫|X(jω)|2dω
)1
2
≥ 12
The first term measures the “width” of x(t) around t = 0.
It is like σ if |x(t)|2 was a zero-mean probability distribution.
![Page 81: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/81.jpg)
Uncertainty principle
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 12 / 14
CTFT uncertainty principle:( ∫
t2|x(t)|2dt∫|x(t)|2dt
)1
2
( ∫ω2|X(jω)|2dω∫|X(jω)|2dω
)1
2
≥ 12
The first term measures the “width” of x(t) around t = 0.
It is like σ if |x(t)|2 was a zero-mean probability distribution.The second term is similarly the “width” of X(jω) in frequency.
![Page 82: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/82.jpg)
Uncertainty principle
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 12 / 14
CTFT uncertainty principle:( ∫
t2|x(t)|2dt∫|x(t)|2dt
)1
2
( ∫ω2|X(jω)|2dω∫|X(jω)|2dω
)1
2
≥ 12
The first term measures the “width” of x(t) around t = 0.
It is like σ if |x(t)|2 was a zero-mean probability distribution.The second term is similarly the “width” of X(jω) in frequency.
A signal cannot be concentrated in both time and frequency.
![Page 83: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/83.jpg)
Uncertainty principle
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 12 / 14
CTFT uncertainty principle:( ∫
t2|x(t)|2dt∫|x(t)|2dt
)1
2
( ∫ω2|X(jω)|2dω∫|X(jω)|2dω
)1
2
≥ 12
The first term measures the “width” of x(t) around t = 0.
It is like σ if |x(t)|2 was a zero-mean probability distribution.The second term is similarly the “width” of X(jω) in frequency.
A signal cannot be concentrated in both time and frequency.
Proof Outline:Assume
∫
|x(t)|2dt = 1⇒
∫
|X(jω)|2dω = 2π [Parseval]
![Page 84: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/84.jpg)
Uncertainty principle
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 12 / 14
CTFT uncertainty principle:( ∫
t2|x(t)|2dt∫|x(t)|2dt
)1
2
( ∫ω2|X(jω)|2dω∫|X(jω)|2dω
)1
2
≥ 12
The first term measures the “width” of x(t) around t = 0.
It is like σ if |x(t)|2 was a zero-mean probability distribution.The second term is similarly the “width” of X(jω) in frequency.
A signal cannot be concentrated in both time and frequency.
Proof Outline:Assume
∫
|x(t)|2dt = 1⇒
∫
|X(jω)|2dω = 2π [Parseval]
Set v(t) = dxdt⇒ V (jω) = jωX(jω) [by parts]
![Page 85: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/85.jpg)
Uncertainty principle
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 12 / 14
CTFT uncertainty principle:( ∫
t2|x(t)|2dt∫|x(t)|2dt
)1
2
( ∫ω2|X(jω)|2dω∫|X(jω)|2dω
)1
2
≥ 12
The first term measures the “width” of x(t) around t = 0.
It is like σ if |x(t)|2 was a zero-mean probability distribution.The second term is similarly the “width” of X(jω) in frequency.
A signal cannot be concentrated in both time and frequency.
Proof Outline:Assume
∫
|x(t)|2dt = 1⇒
∫
|X(jω)|2dω = 2π [Parseval]
Set v(t) = dxdt⇒ V (jω) = jωX(jω) [by parts]
Now∫
txdxdtdt= 1
2 tx2(t)
∣
∣
∞t=−∞ −
∫
12x
2dt = 0− 12 [by parts]
![Page 86: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/86.jpg)
Uncertainty principle
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 12 / 14
CTFT uncertainty principle:( ∫
t2|x(t)|2dt∫|x(t)|2dt
)1
2
( ∫ω2|X(jω)|2dω∫|X(jω)|2dω
)1
2
≥ 12
The first term measures the “width” of x(t) around t = 0.
It is like σ if |x(t)|2 was a zero-mean probability distribution.The second term is similarly the “width” of X(jω) in frequency.
A signal cannot be concentrated in both time and frequency.
Proof Outline:Assume
∫
|x(t)|2dt = 1⇒
∫
|X(jω)|2dω = 2π [Parseval]
Set v(t) = dxdt⇒ V (jω) = jωX(jω) [by parts]
Now∫
txdxdtdt= 1
2 tx2(t)
∣
∣
∞t=−∞ −
∫
12x
2dt = 0− 12 [by parts]
So 14 =
∣
∣
∫
txdxdtdt∣
∣
2≤
(∫
t2x2dt)
(
∫∣
∣
dxdt
∣
∣
2dt)
[Schwartz]
![Page 87: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/87.jpg)
Uncertainty principle
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 12 / 14
CTFT uncertainty principle:( ∫
t2|x(t)|2dt∫|x(t)|2dt
)1
2
( ∫ω2|X(jω)|2dω∫|X(jω)|2dω
)1
2
≥ 12
The first term measures the “width” of x(t) around t = 0.
It is like σ if |x(t)|2 was a zero-mean probability distribution.The second term is similarly the “width” of X(jω) in frequency.
A signal cannot be concentrated in both time and frequency.
Proof Outline:Assume
∫
|x(t)|2dt = 1⇒
∫
|X(jω)|2dω = 2π [Parseval]
Set v(t) = dxdt⇒ V (jω) = jωX(jω) [by parts]
Now∫
txdxdtdt= 1
2 tx2(t)
∣
∣
∞t=−∞ −
∫
12x
2dt = 0− 12 [by parts]
So 14 =
∣
∣
∫
txdxdtdt∣
∣
2≤
(∫
t2x2dt)
(
∫∣
∣
dxdt
∣
∣
2dt)
[Schwartz]
=(∫
t2x2dt)
(
∫
|v(t)|2 dt)
![Page 88: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/88.jpg)
Uncertainty principle
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 12 / 14
CTFT uncertainty principle:( ∫
t2|x(t)|2dt∫|x(t)|2dt
)1
2
( ∫ω2|X(jω)|2dω∫|X(jω)|2dω
)1
2
≥ 12
The first term measures the “width” of x(t) around t = 0.
It is like σ if |x(t)|2 was a zero-mean probability distribution.The second term is similarly the “width” of X(jω) in frequency.
A signal cannot be concentrated in both time and frequency.
Proof Outline:Assume
∫
|x(t)|2dt = 1⇒
∫
|X(jω)|2dω = 2π [Parseval]
Set v(t) = dxdt⇒ V (jω) = jωX(jω) [by parts]
Now∫
txdxdtdt= 1
2 tx2(t)
∣
∣
∞t=−∞ −
∫
12x
2dt = 0− 12 [by parts]
So 14 =
∣
∣
∫
txdxdtdt∣
∣
2≤
(∫
t2x2dt)
(
∫∣
∣
dxdt
∣
∣
2dt)
[Schwartz]
=(∫
t2x2dt)
(
∫
|v(t)|2 dt)
=(∫
t2x2dt)
(
12π
∫
|V (jω)|2 dω)
![Page 89: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/89.jpg)
Uncertainty principle
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 12 / 14
CTFT uncertainty principle:( ∫
t2|x(t)|2dt∫|x(t)|2dt
)1
2
( ∫ω2|X(jω)|2dω∫|X(jω)|2dω
)1
2
≥ 12
The first term measures the “width” of x(t) around t = 0.
It is like σ if |x(t)|2 was a zero-mean probability distribution.The second term is similarly the “width” of X(jω) in frequency.
A signal cannot be concentrated in both time and frequency.
Proof Outline:Assume
∫
|x(t)|2dt = 1⇒
∫
|X(jω)|2dω = 2π [Parseval]
Set v(t) = dxdt⇒ V (jω) = jωX(jω) [by parts]
Now∫
txdxdtdt= 1
2 tx2(t)
∣
∣
∞t=−∞ −
∫
12x
2dt = 0− 12 [by parts]
So 14 =
∣
∣
∫
txdxdtdt∣
∣
2≤
(∫
t2x2dt)
(
∫∣
∣
dxdt
∣
∣
2dt)
[Schwartz]
=(∫
t2x2dt)
(
∫
|v(t)|2 dt)
=(∫
t2x2dt)
(
12π
∫
|V (jω)|2 dω)
=(∫
t2x2dt)
(
12π
∫
ω2 |X(jω)|2dω
)
![Page 90: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/90.jpg)
Uncertainty principle
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 12 / 14
CTFT uncertainty principle:( ∫
t2|x(t)|2dt∫|x(t)|2dt
)1
2
( ∫ω2|X(jω)|2dω∫|X(jω)|2dω
)1
2
≥ 12
The first term measures the “width” of x(t) around t = 0.
It is like σ if |x(t)|2 was a zero-mean probability distribution.The second term is similarly the “width” of X(jω) in frequency.
A signal cannot be concentrated in both time and frequency.
Proof Outline:Assume
∫
|x(t)|2dt = 1⇒
∫
|X(jω)|2dω = 2π [Parseval]
Set v(t) = dxdt⇒ V (jω) = jωX(jω) [by parts]
Now∫
txdxdtdt= 1
2 tx2(t)
∣
∣
∞t=−∞ −
∫
12x
2dt = 0− 12 [by parts]
So 14 =
∣
∣
∫
txdxdtdt∣
∣
2≤
(∫
t2x2dt)
(
∫∣
∣
dxdt
∣
∣
2dt)
[Schwartz]
=(∫
t2x2dt)
(
∫
|v(t)|2 dt)
=(∫
t2x2dt)
(
12π
∫
|V (jω)|2 dω)
=(∫
t2x2dt)
(
12π
∫
ω2 |X(jω)|2dω
)
No exact equivalent for DTFT/DFT but a similar effect is true
![Page 91: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/91.jpg)
Summary
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 13 / 14
• Three types: CTFT, DTFT, DFT
![Page 92: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/92.jpg)
Summary
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 13 / 14
• Three types: CTFT, DTFT, DFT
DTFT = CTFT of continuous signal × impulse train
![Page 93: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/93.jpg)
Summary
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 13 / 14
• Three types: CTFT, DTFT, DFT
DTFT = CTFT of continuous signal × impulse train
DFT = DTFT of periodic or finite support signal
![Page 94: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/94.jpg)
Summary
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 13 / 14
• Three types: CTFT, DTFT, DFT
DTFT = CTFT of continuous signal × impulse train
DFT = DTFT of periodic or finite support signal
• DFT is a scaled unitary transform
![Page 95: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/95.jpg)
Summary
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 13 / 14
• Three types: CTFT, DTFT, DFT
DTFT = CTFT of continuous signal × impulse train
DFT = DTFT of periodic or finite support signal
• DFT is a scaled unitary transform
• DTFT: Convolution → Product; Product → Circular Convolution
![Page 96: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/96.jpg)
Summary
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 13 / 14
• Three types: CTFT, DTFT, DFT
DTFT = CTFT of continuous signal × impulse train
DFT = DTFT of periodic or finite support signal
• DFT is a scaled unitary transform
• DTFT: Convolution → Product; Product → Circular Convolution
• DFT: Product ↔ Circular Convolution
![Page 97: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/97.jpg)
Summary
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 13 / 14
• Three types: CTFT, DTFT, DFT
DTFT = CTFT of continuous signal × impulse train
DFT = DTFT of periodic or finite support signal
• DFT is a scaled unitary transform
• DTFT: Convolution → Product; Product → Circular Convolution
• DFT: Product ↔ Circular Convolution
• DFT: Zero Padding → Denser freq sampling but same resolution
![Page 98: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/98.jpg)
Summary
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 13 / 14
• Three types: CTFT, DTFT, DFT
DTFT = CTFT of continuous signal × impulse train
DFT = DTFT of periodic or finite support signal
• DFT is a scaled unitary transform
• DTFT: Convolution → Product; Product → Circular Convolution
• DFT: Product ↔ Circular Convolution
• DFT: Zero Padding → Denser freq sampling but same resolution
• Phase is only defined to within a multiple of 2π.
![Page 99: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/99.jpg)
Summary
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 13 / 14
• Three types: CTFT, DTFT, DFT
DTFT = CTFT of continuous signal × impulse train
DFT = DTFT of periodic or finite support signal
• DFT is a scaled unitary transform
• DTFT: Convolution → Product; Product → Circular Convolution
• DFT: Product ↔ Circular Convolution
• DFT: Zero Padding → Denser freq sampling but same resolution
• Phase is only defined to within a multiple of 2π.
• Whenever you integrate over frequency you need a scale factor
12π for CTFT and DTFT or 1
Nfor DFT
e.g. Inverse transform, Parseval, frequency domain convolution
![Page 100: 2: Three Different Fourier Transforms](https://reader036.vdocuments.site/reader036/viewer/2022090905/613c8d1da9aa48668d4a3e3e/html5/thumbnails/100.jpg)
Summary
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 13 / 14
• Three types: CTFT, DTFT, DFT
DTFT = CTFT of continuous signal × impulse train
DFT = DTFT of periodic or finite support signal
• DFT is a scaled unitary transform
• DTFT: Convolution → Product; Product → Circular Convolution
• DFT: Product ↔ Circular Convolution
• DFT: Zero Padding → Denser freq sampling but same resolution
• Phase is only defined to within a multiple of 2π.
• Whenever you integrate over frequency you need a scale factor
12π for CTFT and DTFT or 1
Nfor DFT
e.g. Inverse transform, Parseval, frequency domain convolution
For further details see Mitra: 3 & 5.
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MATLAB routines
2: Three Different FourierTransforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Uncertainty principle
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10159) Fourier Transforms: 2 – 14 / 14
fft, ifft DFT with optional zero-paddingfftshift swap the two halves of a vectorconv convolution or polynomial multiplication (not circular)
x[n]⊛y[n] real(ifft(fft(x).*fft(y)))unwrap remove 2π jumps from phase spectrum