Continuous-Time Signal
Analysis: The Fourier Transform
Chapter 7
Mohamed Bingabr
Chapter Outline
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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 ejn t.
o
Fourier transform (FT) allows us to represent
aperiodic (not periodic) signal in term of exponentials
ejt.
xTo(t)
xT0 t  
1
Dn 
T0

jn0t
D
e
 n
n  
T0 / 2
 jn0 t
x
(
t
)
e
 T0
T0 / 2
Link between FT and FS xT(t)
xTo(t)
T0    0  0
xt   lim xT0 t 
T0 
As T0 gets larger and larger the fundamental frequency 0 gets
smaller and smaller so the spectrum becomes continuous.
Dn
1
Dn  X (n0 )
T0
0
X ()

The Fourier Transform Spectrum
The Fourier transform:
X ( ) 

 j t
x
(
t
)
e
dt


X ( )  X ( ) e X ( )
The Amplitude (Magnitude) Spectrum
The Phase Spectrum
The amplitude spectrum is an even function and the phase is an
odd function.
The Inverse Fourier transform:
1
x(t ) 
2

 X ()e

jt
d
Example
Find the Fourier transform of x(t) = e-atu(t), the
magnitude, and the spectrum
Solution:
S-plane

1
 at  jt
s =  + j
X ( )   e e dt 
if a  0
j
a

j

0
X ( ) 
1
a2   2
X ( )   tan1 ( / a)
-a
How does X() relates to X(s)?

1 ( s  a ) t
X ( s)   e e dt  
e
as
0
Re(s)


ROC
 at  st
0
1
X ( s) 
if Re(s)  -a
as
Since the j-axis is in the region of convergence then FT exist.
Useful Functions
Unit Gate Function
0
 x 
rect   0.5
  
1
| x |  / 2
| x |  / 2
| x |  / 2
1
-/2
/2
x
/2
x
Unit Triangle Function
 x  0
   
   1  2 x / 
| x |  / 2
| x |  / 2
1
-/2
Useful Functions
Interpolation Function
sin x
sinc( x) 
x
sinc( x)  0 for x   k
sinc( x)  1 for x  0
sinc(x)
x
Example
Find the FT, the magnitude, and the phase spectrum
of x(t) = rect(t/).
Answer
X ( ) 
 /2
 rect(t /  )e
 jt
dt   sinc( / 2)
 /2
What is the bandwidth of the above pulse?
The spectrum of a pulse extend from 0 to . However, much of
the spectrum is concentrated within the first lobe (=0 to 2/)
Examples
Find the FT of the unit impulse (t).
Answer

X ( )    (t )e jt dt  1

Find the inverse FT of ().
Answer
1
x(t ) 
2

1
 ( )e d  2
jt
so thespectrumof a constantis an impulse
1  2 ( )
Examples
Find the inverse FT of (- 0).
Answer

1
1 j 0 t
jt
x(t ) 
 (  0 )e d 
e

2 
2
so thespectrumof a complexexponentis a shiftedimpulse
e j0t  2 (  0 )
and
e  j0t  2 (  0 )
Find the FT of the everlasting sinusoid cos(0t).
Answer
1 j 0 t
cos0t  e  e  j0t
2
1 j 0 t
e  e  j0t    (  0 )   (  0 )
2




Examples
Find the FT of a periodic signal.
Answer
x(t ) 
n 
D e
n  
jn0t
n
0  2 / T0
T aketheFT of both side and use linearitypropertyof FT
X ( )  2
n 
 D  (  n )
n  
n
0
Examples
Find the FT of the unit impulse train T0 (t )
Answer
1
 T0 (t ) 
T0
n 
e
jn0t
n  
2
X ( ) 
T0
n 
  (  n )
n  
0
Properties of the Fourier Transform
• Linearity:
• Let
then
xt   X  
and
yt   Y  
xt   yt   X    Y  
• Time Scaling:
• Let
then
xt   X  
1  
xat  X  
a a
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:
• Let
then
xt   X  
x(t )  X ( )
Example: Find the FT of eatu(-t) and e-a|t|
• Left or Right Shift in Time:
• Let
then
xt   X  
Time shift effects the
phase and not the
magnitude.
xt  t0   X  e
 jt0
Example: if x(t) = sin(t) then what is the FT of x(t-t0)?
Example: Find the FT of
spectrum
e
 a|t t0 |
and draw its magnitude and
Properties of the Fourier Transform
• Multiplication by a Complex Exponential (Freq. Shift
Property):
• Let
then
xt   X  
x(t )e
j0t
 X (  0 )
• Multiplication by a Sinusoid (Amplitude Modulation):
Let
then
xt   X  
1
xt  cos 0t   X   0   X   0 
2
cos0t is the carrier, x(t) is the modulating signal (message),
x(t) cos0t is the modulated signal.
Example: Amplitude Modulation
x(t)
Example: Find the FT for the signal
x(t )  rect(t / 4) cos10t
A
-2
HW10_Ch7: 7.1-1, 7.1-5, 7.1-6, 7.2-1, 7.2-2, 7.2-4, 7.3-2
2
Amplitude Modulation
Modulation
 AM (t )  m(t ) cosct
Demodulation
 AM (t ) cosct  0.5m(t )[1  cos2ct ]
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 Frequency Domain:
• Let
then
xt   X  
n
d
n
n
t x(t )  ( j )
X ( )
n
d
• Differentiation in the Time Domain:
Let
then
xt   X  
n
d
n
x(t )  ( j ) X ( )
n
dt
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:
Let
xt   X  
t
Then
1
 x( )d  j X ( )   X (0) ( )
• Convolution and Multiplication in the Time Domain:
Let
xt   X  
yt   Y  
Then
x(t )  y(t )  X ( )Y ( )
1
x1 (t ) x2 (t ) 
X 1 ( )  X 2 ( )
2
Frequency convolution
Example
Find 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 Theorem: since x(t) is non-periodic
and has FT X(), then it is an energy signals:

1
E   xt  dt 
2

2

 X  
2
d

Real signal has even spectrum X()= X(-), E 

X  


1
2
d
0
Example
Find 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) :
• Let
then
xt   X  
X (t )  2 x( )
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 Functions
1- Truncate x(t) to reduce numerical computation
2- Truncate h(t) to make the system response finite and causal
3- Truncate X() to prevent aliasing in sampling the signal x(t)
4- Truncate Dn to synthesis the signal x(t) from few harmonics.
xw (t )  x(t ) w(t )
and
1
X w ( ) 
X ( ) *W ( )
2
What are the implications of data truncation?
Implications of Data Truncation
1- Spectral spreading
2- Poor frequency resolution
3- 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 Theorem
A real signal whose spectrum is bandlimited to B Hz [X()=0 for
|| >2B ] 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 
x (t )  x(nT )  x(t )   (t  nT )
n  
1 n  jns t
x (t )  x(nT )  x(t )  e
T n  
1 n 
X ( )   X (  ns )
T n  
Reconstructing the Signal from the Samples
X ( )  H ( ) X ( )
x(t )  h(t ) * x(nT )
x(t )  h(t ) *  x(nT ) (t  nT )
n
x(t )   x(nT )h(t  nT )
n
x(t )   x(nT )sinc(2B(t  nT )
n
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 Hz
The Nyquist rate = 2 fmax = 2*15 = 30 sample/sec
Aliasing
If 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.
Frequency above Fs/2
will appear (aliased) as
frequency below Fs/2
LPF
With cutoff
frequency
Fs/2
Quantization & Binary Representation
x(t)
L2
n
4
3
2
1
0
-1
-2
-3
L : number of levels
n : Number of bits
Quantization error = x/2
xmax  xmin
x 
L 1
4
3
2
1
0
-1
-2
-3
x
111
110
101
100
011
010
001
000
111
110
101
100
011
010
001
000
Example
A 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 8.3-4
Five 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
X ( ) 

 jt
x
(
t
)
e
dt


1 
X ( )   x(n)e  jn
T n 
N -1
X (k )   x(n)e  j 2kn / N
n 0
k
Link between Continuous and Discrete
x(t) Sampling Theorem
x(n)
x(t)
x(n)
t
n
Continuous
x(t) Laplace Transform

Discrete
X(s)
x(n)
X (s)   x(t )est dt
z Transform
X ( z) 
x(t)
X ( ) 

 x(t )e

 jt
X(j)
dt
n
x
(
n
)
z

X(z)
n  

Fourier Transform
n 
Discrete Fourier Transform
x(n)
N-1
X (k )   x(n)e
n 0
j
2k
n
N
X(k)