#### Transcript Continuous-Time Signal Analysis: The Fourier Transform

```ENGR 4323/5323
Digital and Analog Communication
Ch 3
Analysis and Transmission of Signals
Engineering and Physics
University of Central Oklahoma
Dr. 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 Distortion over a Communication Channel
Signal Energy and Energy Spectral Density
Signal Power and Power Spectral Density
The Fourier Transform Spectrum
Fourier transform (FT) allows us to represent
aperiodic (not periodic) signal in term of its frequency
ω.
x(t)
The Fourier transform integrals:
X ( ) 

 j t
x
(
t
)
e
dt


X ( )  X ( ) e X ( )
|X(ω)|
ω
The Fourier Transform Spectrum
X ( )
X ()  X () e
The Phase Spectrum
The Amplitude (Magnitude) Spectrum
The amplitude spectrum is an even function and the phase is an
odd function.
The Inverse Fourier transform:
1
x(t ) 
2

jt
X
(

)
e
d


Useful Functions
Unit Gate Function
t
rect 

0
 
  0.5
 
1
| t |  / 2
| t |  / 2
| t |  / 2
1
-/2
/2
t
/2
t
Unit Triangle Function
t


 0

 1  2 t / 
| t |  / 2
| t |  / 2
1
-/2
Useful Functions
Interpolation Function
sin t
sinc(t ) 
t
sinc(t )  0 for t   k
sinc(t )  1 for t  0
sinc(t)
t
Example
Find the FT, the magnitude, and the phase spectrum
of x(t) = rect(t/).
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).

X ( )    (t )e jt dt  1

Find the inverse FT of ().
1
x(t ) 
2

1
 ( )e d  2
jt
so thespectrumof a constantis an impulse
1  2 ( )
Examples
Find the FT of the unit impulse train T (t )
0
1
 T0 (t ) 
T0
n 
jn0t
e

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 ( )
• Left or Right Shift in Time:
• Let
xt   X  
then
xt  t0   X  e jt0
Time shift effects the
phase and not the
magnitude.
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
A
x(t )  rect(t / 4) cos10t
-2
2
Amplitude Modulation
Modulation
 AM (t )  m(t ) cosct
Demodulation
 AM (t ) cos 2 c t  0.5m(t )[1  cos 2c t ]
Then lowpass filtering
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 
1



X   d
2
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.
1
1
𝑥
−1
dx = 𝑡𝑎𝑛
𝑎2 + 𝑥 2
𝑎
𝑎
Properties of the Fourier Transform
• Duality ( Similarity) :
• Let
then
xt   X  
X (t )  2 x( )
Signal Transmission Through a
Linear System
Distortionless Transmission (System)
Slope is constant for
distortionless system
Example 3.16
A transmission medium is modeled by a simple RC low-pass
filter shown below. If g(t) and y(t) are the input and the output,
respectively to the circuit, determine the transfer function H(f),
θh(f), and td(f). For distortionless transmission through this filter,
what is the requirement on the bandwidth of g(t) if amplitude
response variation within 2% and time delay variation within 5%
are tolerable? What is the transmission delay? Find the output
y(t).
𝑑
𝑎
−1
𝑡𝑎𝑛 𝑎𝑥 =
𝑑𝑥
1 + 𝑎2 𝑥 2
Ideal Versus Practical Filters
wR(t)
t
H(ω)*WR(t)
h(t)wR(t)
Ideal Versus Practical Filters
Signal Distortion Over a
Communication Channel
1. Linear Distortion
2. Channel Nonlinearities
3. Multipath Effects
- Channel fading vary with time. To overcome this
distortion is to use automatic gain control (AGC)
Linear Distortion
Channel causes magnitude distortion, phase distortion, or both.
Example: A channel is modeled by a low-pass filter with transfer
function H(f) give by
−𝑗2𝜋𝑓𝑡𝑑
1
+
𝑘𝑐𝑜𝑠2𝜋𝑓𝑇
𝑒
𝐻(𝑓) =
0
𝑓 <𝐵
𝑓 >𝐵
A pulse g(t) band-limited to B Hz is applied at the input of this
filter. Find the output y(t).
Nonlinear Distortion
y(t) = f(g(t))
f(g) can be expanded by Maclaurin series
y 𝑡 = 𝑎0 + 𝑎1 𝑔 𝑡 + 𝑎2 𝑔2 𝑡 + ⋯ + 𝑎𝑘 𝑔𝑘 𝑡
If the bandwidth of g(t) is B Hz then the bandwidth of y(t) is kB Hz.
Example: The input x(t) and the output y(t) of a certain nonlinear
channel are related as
y(t) = x(t) + 0.000158 x2(t)
Find the output signal y(t) and its spectrum Y(f) if the input
signal is x(t) = 2000 sinc(2000t). Verify that the bandwidth of the
output signal is twice that of the input signal. This is the result of
signal squaring. Can the signal x(t) be recovered (without
distortion) from the output y(t)?
Continue Example
Distortion Caused by Multipath Effects
𝐻 𝑓 = 𝑒 −𝑗2𝜋𝑓𝑡𝑑 + α𝑒 −𝑗2𝜋𝑓(𝑡𝑑 +∆𝑡)
𝐻 𝑓 = 𝑒 −𝑗2𝜋𝑓𝑡𝑑 (1 + α𝑒 −𝑗2𝜋𝑓∆𝑡 )
𝐻 𝑓 = 𝑒 −𝑗2𝜋𝑓𝑡𝑑 (1 + α 𝑐𝑜𝑠2𝜋𝑓∆𝑡 − 𝑗α 𝑠𝑖𝑛2𝜋𝑓∆𝑡)
𝐻 𝑓 =
1+
α2
+ 2α 𝑐𝑜𝑠2𝜋𝑓∆𝑡 𝑒𝑥𝑝 −𝑗 2𝜋𝑓𝑡𝑑 +
𝑡𝑎𝑛−1
α 𝑠𝑖𝑛2𝜋𝑓∆𝑡
1 + α 𝑐𝑜𝑠2𝜋𝑓∆𝑡
Common distortion in this type of channel is frequency selective
Energy and Energy Spectral Density
∞
𝐸𝑔 =
𝑔 𝑡 𝑔∗ 𝑡 𝑑𝑡
Energy in the time domain
𝐺(𝑓) 2 𝑑𝑓
Energy in the frequency domain
−∞
∞
𝐸𝑔 =
−∞
Energy spectral density (ESD), Ψ𝑔 (𝑓), is the energy per unit
bandwidth (in hertz) of the spectral components of g(t) centered
at frequency f.
Ψ𝑔 (𝑓) = 𝐺(𝑓)
2
The ESD of the system’s output in term of the input ESD is
Ψ𝑥 (𝑓)
𝐻(𝑓)
Ψ𝑦 (𝑓) = 𝐻(𝑓) 2 Ψ𝑥 (𝑓)
Essential Bandwidth of a Signal
Estimate the essential bandwidth of a rectangular pulse g(t) =
(t/T), where the essential bandwidth must contain at least 90%
of the pulse energy.
∞
𝐸𝑔 =
−∞
𝐵
𝐸𝐵 =
𝑔2 𝑡 𝑑𝑡 =
𝑇/2
𝑑𝑡 = 𝑇
−𝑇/2
𝑇 2 𝑠𝑖𝑛𝑐 2 𝜋𝑓𝑇 𝑑𝑓 = 0.9𝑇
−𝐵
B = 1/T Hz
Energy of Modulated Signals
The modulated signal appears more energetic than the signal
g(t) but its energy is half of the energy of the signal g(t). Why?
𝜑 𝑡 = 𝑔 𝑡 𝑐𝑜𝑠2𝜋𝑓0 𝑡
1
Φ(𝑓) = 𝐺 𝑓 + 𝑓0 + 𝐺(𝑓 − 𝑓0 )
2
1
Ψ𝜑 (𝑓) = 𝐺 𝑓 + 𝑓0 + 𝐺(𝑓 − 𝑓0 ) 2
4
If f0 > 2B then
1
1
Ψ𝜑 𝑓 = Ψ𝑔 𝑓 + 𝑓0 + Ψ𝑔 (𝑓 + 𝑓0 )
4
4
1
𝐸𝜑 = 𝐸𝑔
2
Time Autocorrelation Function and
Energy Spectral Density
The autocorrelation 𝜓𝑔 (𝜏) of a signal g(t) and its ESD Ψ𝑔 (𝑓)
form a Fourier transform pair, that is
𝜓𝑔 𝜏
𝐹𝑇 𝑎𝑛𝑑 𝐼𝐹𝑇
Ψ𝑔 (𝑓)
Example: Find the time autocorrelation function of the signal
g(t) = e-atu(t), and from it determine the ESD of g(t).
Signal Power and Power Spectral
Density
Power Pg of the signal g(t)
1
𝑃𝑔 = lim
𝑇→∞ 𝑇
𝑇/2
𝑔 𝑡 𝑔∗ 𝑡 𝑑𝑡
−𝑇/2
𝐸𝑔𝑇
𝑃𝑔 = lim
𝑇→∞ 𝑇
Power spectral density Sg(f) of the signal g(t)
𝐺𝑇 (𝑓)
𝑆𝑔 (𝑓) = lim
𝑇→∞
𝑇
∞
2
𝑃𝑔 =
∞
𝑆𝑔 𝑓 𝑑𝑓 = 2
−∞
𝑆𝑥 (𝑓)
𝐻(𝑓)
𝑆𝑦 (𝑓) = 𝐻(𝑓) 2 𝑆𝑥 (𝑓)
𝑆𝑔 𝑓 𝑑𝑓
0
Time Autocorrelation Function of
Power Signals
Time autocorrelation Rg( ) of a power signal g(t)
1
ℛ𝑔 (𝜏) = lim
𝑇→∞ 𝑇
1
ℛ𝑔 (𝜏) = lim
𝑇→∞ 𝑇
𝑇/2
𝑔 𝑡 𝑔(𝑡 − 𝜏)𝑑𝑡
−𝑇/2
∞
𝑔𝑇 (𝑡)𝑔𝑇 (𝑡 + 𝜏)𝑑𝑡
−∞
𝜓𝑔𝑇 (𝜏)
ℛ𝑔 (𝜏) = lim
𝑇→∞
𝑇
ℛ𝑔 (𝜏)
𝐹𝑇 𝑎𝑛𝑑 𝐼𝐹𝑇
𝑆𝑔 (𝑓)
Autocorrelation a Powerful Tool
If the energy or power spectral density can be found by
the Fourier transform of the signal g(t) then why do we
need to find the time autocorrelation?
Ans: In communication field and in general the signal
g(t) is not deterministic and it is probabilistic function.
Example
A random binary pulse train g(t). The pulse width is Tb/2,
and one binary digit is transmitted every Tb seconds. A
binary 1 is transmitted by positive pulse, and a binary 0
is transmitted by negative pulse. The two symbols are
equally likely and occur randomly. Determine the PSD
and the essential bandwidth of this signal.
1
1
0
1
0
0
1
0
Challenge: g(t) is not deterministic and can not be
expressed mathematically to find the Fourier transform
and PSD. g(t) is random signal.
1
ℛ𝑔 (𝜏) = lim
𝑇→∞ 𝑇
𝑇/2
𝑔 𝑡 𝑔(𝑡 − 𝜏)𝑑𝑡
−𝑇/2
Tb
g(t)
t
𝜏
0
For 0 < 𝜏 < 𝑇𝑏 /2
1 𝑇𝑏
1 𝜏
ℛ𝑔 𝜏 = lim
−𝜏 𝑁 =
−
𝑁→∞ 𝑁𝑇𝑏 2
2 𝑇𝑏
For 𝜏 > 𝑇𝑏 /2
ℛ𝑔 𝜏 = 0
g(t-τ)
Homework Problem
𝑇𝑏 /2 < 𝜏 < 𝑇𝑏
(𝑘 + 1)𝑇𝑏
𝜏
𝑘𝑇𝑏 + 𝜏 + 𝑇𝑏 /2
𝑘𝑇𝑏
𝑘𝑇𝑏 + 𝜏
1
𝑇𝑏
𝑁
ℛ𝑔 𝜏 = lim
𝑘𝑇𝑏 + 𝜏 + − (𝑘 + 1)𝑇𝑏
𝑁→∞ 𝑁𝑇𝑏
2
4
1 𝜏 1
ℛ𝑔 𝜏 =
−
4 𝑇𝑏 2
ℛ𝑔 𝜏
1/8
−𝑇𝑏 −𝑇𝑏 /2
𝑇𝑏 /2
𝑇𝑏
Discrete Fourier Transform (DFT, FFT)
𝑇0
𝑁0 =
𝑇𝑠
2𝜋
Ω0 =
𝑁0
𝑁0 −1
𝑔𝑘 𝑒 −𝑗𝑞Ω0 𝑘
𝐺𝑞 =
𝑘=0
```