continuous-time signal analysis: the fourier transform chapter 7 mohamed bingabr
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Continuous-Time Signal Analysis: The Fourier Transform
Chapter 7
Mohamed Bingabr
Chapter Outline
• Aperiodic Signal Representation by Fourier Integral• Fourier Transform of Useful Functions• Properties of Fourier Transform• Signal Transmission Through LTIC Systems• Ideal and Practical Filters• Signal Energy• Applications to Communications• Data Truncation: Window Functions
Link between FT and FS
Fourier series (FS) allows us to represent periodic signal in term of sinusoidal or exponentials e jnot.
Fourier transform (FT) allows us to represent aperiodic (not periodic) signal in term of exponentials ejt.
xTo(t)
n
tjnnT eDtx 0
0
tjnT
T
Tn etxT
D 0
0
0
0
2/
2/0
)(1
Link between FT and FS
txtx TT 0
0
lim
000 T
xTo(t)xT(t)
nD )(X
As T0 gets larger and larger the fundamental frequency 0 gets smaller and smaller so the spectrum becomes continuous.
0
)(1
00
nXT
Dn
The Fourier Transform Spectrum
The Inverse Fourier transform:
deXtx tj)(2
1)(
The Fourier transform:
)()()(
)()(
X
tj
eXX
dtetxX
The Amplitude (Magnitude) Spectrum The Phase Spectrum
The amplitude spectrum is an even function and the phase is an odd function.
ExampleFind the Fourier transform of x(t) = e-atu(t), the magnitude, and the spectrumSolution:
)/(tan)(1
)(
0a if 1
)(
1
22
0
aXa
X
jadteeX tjat
How does X() relates to X(s)?
-aRe(s) if 1
)(
1
)(0
)(
0
sasX
esa
dteesX tasstat
S-planes = + j
Re(s)
j
ROC
-a
Since the j-axis is in the region of convergence then FT exist.
Useful FunctionsUnit Gate Function
2/|| 1
2/|| 5.0
2/|| 0
x
x
xx
rect
Unit Triangle Function
2/|| /21
2/|| 0
xx
xx
/2-/2
/2-/2
1
1
x
x
Useful FunctionsInterpolation Function
0for x 1)(sinc
for x 0)(sinc
sin)(sinc
x
kxx
xx
sinc(x)
x
ExampleFind the FT, the magnitude, and the phase spectrum of x(t) = rect(t/).
Answer
)2/sinc()/()(2/
2/
dtetrectX tj
The spectrum of a pulse extend from 0 to . However, much of the spectrum is concentrated within the first lobe (=0 to 2/)
What is the bandwidth of the above pulse?
ExamplesFind the FT of the unit impulse (t).Answer
1)()(
dtetX tj
Find the inverse FT of ().Answer
)(21
impulsean isconstant a of spectrum theso
2
1)(
2
1)(
detx tj
ExamplesFind the inverse FT of (- 0).Answer
)(2 and )(2
impulse shifted a isexponent complex a of spectrum theso
2
1)(
2
1)(
00
0
00
0
tjtj
tjtj
ee
edetx
Find the FT of the everlasting sinusoid cos(0t).Answer
)()(2
12
1cos
00
0
00
00
tjtj
tjtj
ee
eet
ExamplesFind the FT of a periodic signal.Answer
n
nn
tjnn
nn
nDX
TeDtx
)(2)(
FT ofproperty linearity use and sideboth of FT theTake
/2)(
0
000
ExamplesFind the FT of the unit impulse train Answer
)(0tT
n
n
n
n
tjnT
nT
X
eT
t
)(2
)(
1)(
00
0
0
0
Properties of the Fourier Transform• Linearity:Linearity:
• Let and Let and
thenthen
Xtx Yty
YXtytx
• Time Scaling:Time Scaling:
• LetLet
thenthen
Xtx
a
Xa
atx1
Compression in the time domain results in expansion in the frequency domain
Internet channel A can transmit 100k pulse/sec and channel B can transmit 200k pulse/sec. Which channel does require higher bandwidth?
Properties of the Fourier Transform• Time Reversal:Time Reversal:
• LetLet
thenthen ( ) ( )x t X Xtx
Example: Find the FT of eatu(-t) and e-a|t|
• Left or Right Shift in Time:Left or Right Shift in Time:
• LetLet
thenthen
Xtx
00
tjeXttx Example: if x(t) = sin(t) then what is the FT of x(t-t0)?
Time shift effects the phase and not the magnitude.
Example: Find the FT of and draw its magnitude and spectrum
|| 0ttae
Properties of the Fourier Transform• Multiplication by a Complex Exponential (Freq. Shift Multiplication by a Complex Exponential (Freq. Shift
Property):Property):
• LetLet
then then 00( ) ( )j tx t e X
Xtx
• Multiplication by a Sinusoid (Amplitude Modulation):Multiplication by a Sinusoid (Amplitude Modulation):
LetLet
thenthen
Xtx
000 2
1cos XXttx
cos0t is the carrier, x(t) is the modulating signal (message),x(t) cos0t is the modulated signal.
Example: Amplitude Modulation
Example: Find the FT for the signal
-2 2
A
x(t)
ttrecttx 10cos)4/()(
HW10_Ch7: 7.1-1, 7.1-5, 7.1-6, 7.2-1, 7.2-2, 7.2-4, 7.3-2
Amplitude Modulation
ttmt cAM cos)()( Modulation
]2cos1)[(5.0 cos)( ttmtt ccAM Demodulation
Then lowpass filtering
Amplitude Modulation: Envelope Detector
Applic. of Modulation: Frequency-Division Multiplexing
1- Transmission of different signals over different bands
2- Require smaller antenna
Properties of the Fourier Transform
• Differentiation in the Time Domain:Differentiation in the Time Domain:
LetLet
thenthen ( ) ( ) ( )n
nn
dx t j X
dt
Xtx
• Differentiation in the Frequency Domain:Differentiation in the Frequency Domain:
• LetLet
thenthen ( ) ( ) ( )n
n nn
dt x t j X
d
Xtx
Example: Use the time-differentiation property to find the Fourier Transform of the triangle pulse x(t) = (t/)
Properties of the Fourier Transform• Integration in the Time Domain:Integration in the Time Domain:
LetLet
ThenThen1
( ) ( ) (0) ( )t
x d X Xj
Xtx
• Convolution and Multiplication in the Time Domain:Convolution and Multiplication in the Time Domain:
LetLet
ThenThen ( ) ( ) ( ) ( )x t y t X Y
Yty
Xtx
)()(2
1)()( 2121
XXtxtx Frequency convolution
ExampleFind the system response to the input x(t) = e-at u(t) if the system impulse response is h(t) = e-bt u(t).
Properties of the Fourier Transform• Parseval’s TheoremParseval’s Theorem: : sincesince xx((tt)) is non-periodicis non-periodic
and has FTand has FT XX(()),, then it is an energy signals:then it is an energy signals:
dXdttxE22
2
1
Real signal has even spectrum XX(())= = XX(-(-)),,
0
21
dXE
ExampleFind the energy of signal x(t) = e-at u(t). Determine the frequency so that the energy contributed by the spectrum components of all frequencies below is 95% of the signal energy EX.
Answer: =12.7a rad/sec
Properties of the Fourier Transform• Duality ( Similarity) :Duality ( Similarity) :
• LetLet
thenthen ( ) 2 ( )X t x
Xtx
HW11_Ch7: 7.3-3(a,b), 7.3-6, 7.3-11, 7.4-1, 7.4-2, 7.4-3, 7.6-1, 7.6-6
Data Truncation: Window Functions1- Truncate x(t) to reduce numerical computation 2- Truncate h(t) to make the system response finite and causal3- Truncate X() to prevent aliasing in sampling the signal x(t)4- Truncate Dn to synthesis the signal x(t) from few harmonics.
What are the implications of data truncation?
)(*)(2
1)( and )()()(
WXXtwtxtx ww
Implications of Data Truncation1- Spectral spreading2- Poor frequency resolution3- Spectral leakage
What happened if x(t) has two spectral components of frequencies differing by less than 4/T rad/s (2/T Hz)?
The ideal window for truncation is the one that has 1- Smaller mainlobe width 2- Sidelobe with high rolloff rate
Data Truncation: Window Functions
Using Windows in Filter Design
=
Using Windows in Filter Design
=
Sampling TheoremA real signal whose spectrum is bandlimited to B Hz [X()=0 for || >2B ] can be reconstructed exactly from its samples taken uniformly at a rate fs > 2B samples per second. When fs= 2B then fs is the Nyquist rate.
n
ns
n
n
tjn
n
n
nXT
X
eT
txnTxtx
nTttxnTxtx
s
)(1
)(
1)()()(
)()()()(
Reconstructing the Signal from the Samples
n
n
n
nTtBnTxtx
nTthnTxtx
nTtnTxthtx
nTxthtx
XHX
)(2(sinc)()(
)()()(
)()(*)()(
)(*)()(
)()()(
LPF
Example
Determine the Nyquist sampling rate for the signal x(t) = 3 + 2 cos(10) + sin(30).
Solution
The highest frequency is fmax = 30/2 = 15 HzThe Nyquist rate = 2 fmax = 2*15 = 30 sample/sec
AliasingIf a continuous time signal is sampled below the Nyquist rate then some of the high frequencies will appear as low frequencies and the original signal can not be recovered from the samples.
LPF With cutoff frequency
Fs/2
Frequency above Fs/2 will appear (aliased) as frequency below Fs/2
Quantization & Binary Representation
111110101100011010001000
111110101100011010001000
43210-1-2-3
43210-1-2-3
nL 2
L : number of levelsn : Number of bitsQuantization error = x/2
x
x(t)
1minmax
L
xxx
ExampleA 5 minutes segment of music sampled at 44000 samples per second. The amplitudes of the samples are quantized to 1024 levels. Determine the size of the segment in bits.
Solution
# of bits per sample = ln(1024) { remember L=2n }n = 10 bits per sample# of bits = 5 * 60 * 44000 * 10 = 13200000 = 13.2 Mbit
Problem 8.3-4Five telemetry signals, each of bandwidth 1 KHz, are quantized and binary coded, These signals are time-division multiplexed (signal bits interleaved). Choose the number of quantization levels so that the maximum error in the peak signal amplitudes is no greater than 0.2% of the peak signal amplitude. The signal must be sampled at least 20% above the Nyquist rate. Determine the data rate (bits per second) of the multiplexed signal.
Discrete-Time Processing of Continuous-Time Signals
Discrete Fourier Transform
dtetxX tj )()(
n
njenxT
X )(1
)(
1-N
0n
/2)()( NknjenxkX
k
Link between Continuous and Discrete
dtetxX tj )()(
1-N
0n
2
)()(n
N
kj
enxkX
x(t) x(n)Sampling Theorem
x(t) Laplace TransformX(s) x(n) X(z)
z Transform
x(t) X(j) x(n) X(k)Fourier Transform Discrete Fourier Transform
dtetxsX st)()(
n
n
nznxzX )()(
t
x(t)
Continuous Discrete
x(n)
n
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