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Discrete-Time Fourier Transform
• Continuous-Time Fourier Transform
• Discrete-Time Fourier Transform.
• Discrete-Time Fourier Transform Theorems.
• Band-Limited Discrete-Time Signals
• The Frequency Response of an LTI Discrete-Time System
• Phase and Group Delays
.)()(
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Continuous-Time Fourier
Transform Pair Equation
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Continuous-Time Fourier
Transform
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Convergence of CTFT
• CTFT of continuous-time signal will only exist if the
Dirichlet conditions are satisfied:-
• (1) Signal must have finite number of discontinuities
and finite number of maximum & minimum in any
finite interval of time.
• (2) Signal must be absolutely integrable:-
∫∞
∞−∞<dttx )(|
Properties of Fourier Transform
• Linearity
• Time Shifting
• Conjugation
• Differentiation in the time-domain
• Integration in the time-domain
• Time and Frequency Scaling.
• Duality
• Parseval’s Relation
• Convolution
• Multiplication
Summary of the approach to
obtain the Fourier Transform for
Discrete-time aperiodic signals.
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Development of Discrete-time
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Example 3.6 Discrete-time Fourier Transform.
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Example 3.6 Discrete-time Fourier Transform.
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Properties of Discrete-time Fourier Transforms.
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Properties of Discrete-time Fourier Transforms.
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−
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Properties of Discrete-time Fourier Transforms.
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↔∴
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Discrete-time Fourier Transform Theorems.
∑ ∫∞
∞=
−
−
=
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Convolution Theorem
h[n]
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x[n]
X(ω)
h[n]*x[n]
H(ω)X(ω)
)()(][*][.
ωω XHnxnhTF
↔
Multiplication Theorem
x[n]h[n]x[n]
)(*)(2
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DTFT
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x
h[n]
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Band-Limited Discrete-Time
Signals. (e.g. Bandpass)
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signal
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plit
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ns
Frequency Response of LTI DTS
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Discrete-Time System
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Frequency Response of LTI IIR
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Concept of Filtering
• Discrete Signals are sequences of numbers in time-domain.
• Through DTFT, Discrete Signals are transformed into frequency domian which are represented as magnitude and phase spectrum of frequencies.
• Shape the Spectrums to the desired forms by multiplying them with frequency response of LTI Discrete-Time Systems.– Frequency shaping filters
– Frequency selective filters.
Frequency Response of LTI DTS
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Bandwidth
||0 0,
|| ,0)(
ππ−pω
In Frequency Domain
Signal Magnitude Spectrum
Magnitude Response
Of LTI DTS (Filter)
cω
What happen to Phase?
• Phase and Group Delays need to be considered
since both DTFT of Input and impulse response
are complex functions.
• Multiplying these two complex functions will
result in addition of their phase components
(argument of complex function)
• Group delay is defined as the negative
derivative of phase function with respect to
angular frequency.
Phase & Group Delay
0
0
j
jjj
jjj
)(- Delay Phase
)(d-Delay Group
input. thelead loutput wil the0,)( IF
input. thelag loutput wil the0,)( IF
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