Digital Signal Processing – Chapter 10
Fourier Analysis of Discrete-Time Signals
and Systems
Dr. Ahmed Samir Fahmy
Associate Professor
Systems and Biomedical Eng. Dept.
Cairo University
Complex Variables
Complex Variables
• Most of the theory of signals and systems is based on
functions of a complex variable
• However, practical signals are functions of a real variable
corresponding to time or space
• Complex variables represent mathematical tools that
allow characteristics of signals to be defined in an easier
to manipulate form
▫ Example: phase of a sinusoidal signal
Complex Numbers and Vectors
• A complex number z represents any point (x, y):
z = x + j y,
▫ x =Re[z] (real part of z)
▫ y =Im[z] (imaginary part of z)
▫ j =Sqrt(-1)
• Vector representation
▫ Rectangular or polar form
▫ Magnitude
and Phase
Complex Numbers and Vectors
• Identical: use either depending on operation
▫ Rectangular form for addition or subtraction
▫ Polar form for multiplication or division
• Example: let
More
Difficult
Complex Numbers and Vectors
• Powers of complex numbers: polar form
• Conjugate
Functions of a Complex Variable
• Just like real-valued functions
▫ Example: Logarithm
• Euler’s identity
▫ Proof: compute polar representation of R.H.S.
▫ Example:
Functions of a Complex Variable
• Starting from Euler’s Identity, one can show:
Phasors and Sinusoidal Steady State
• A sinusoid is a periodic signal represented by,
• If one knows the frequency, cosine is characterized by its
amplitude and phase.
• Define Phasor as complex number characterized by
amplitude and the phase of a cosine signal
▫ Such that
example
• Example: Steady state solution of RC circuit with input
▫ Assume that the steady-state response of this circuit is
also a sinusoid
▫ Then, we can let
▫ Substitute in
The Fourier Transform
On Board Illustration
this slide is intentionally left blank
Discrete-Time Fourier Transform
• Sampled time domain signal  periodic Fourier
transform
• Periodic time domain signal  sampled Fourier
transform
• Periodic and sampled time domain signal  Fourier
transform that is both periodic and sampled
Forward DTFT
Inverse DTFT
Notes on DTFT
DTFT
• DTFT is periodic, with period 2π:
• For convergence:
Notes on DTFT
Eigen functions:
• Let the input to LTI be:
• Using Convolution:
The O/P is similar to
I/P (multiplied by a
factor)
DTFT of h[k]
calculated at input
frequency
DTFT vs. Z-Transform
DTFT
• Compare to z-transform, it is clear that:
• DTFT is the values on the unit circle in z-Transform
• Unit circle must be included in the ROC
DTFT vs. Z-Transform
Example
• Consider the noncausal signal
▫ Determine its DTFT.
▫ Use the obtained DTFT to find
• Solution:
▫ Z-trans:
▫ The ROC is:
1st term:
Overall ROC:
, with
2nd term:
 includes unit circle!
DTFT vs. Z-Transform
Example
• Thus the DTFT is given by
• Also, the DC component is given by:
• Another way to calculate is:
Notes on DTFT
• Problem: the sampled signal is still infinite in length
▫ Impossible to obtain using computers
▫ Consider cases when the signal has compact support
▫ The frequency variable is continuous!
Discrete Fourier Transform (DFT)
• Defined as the Fourier transform of a signal that is both
discrete and periodic
▫ Fourier transform will also be discrete and periodic
▫ Can assume periodicity if we have a finite duration of a
signal: better than zero assumption since it allows
reducing the frequency domain to a sampled function
Discrete Fourier Transform (DFT)
Discrete Fourier Transform (DFT)
Discrete Fourier Transform (DFT)
Discrete Fourier Transform (DFT)
X (e jw )  e jwN1
sin( w( N1  1/ 2))
sin( w / 2)
N 1  j 2 k n
N
X (k )   e
n 0

1  e  j 2 k
1 e
j
2 k
N
N

0
k=0,  2N, ...
o.w.
DFT Formula
• Given a periodic signal x[n] of period N, its DFT is given
by,
• Its inverse is given by,
• Both X[k] and x[n] are periodic of the same period N.
Computation of DFT Using FFT
• A very efficient computation of the DFT is done by
means of the FFT algorithm, which takes advantage of
some special characteristics of the DFT
▫ DFT requires O(N2) computations
▫ FFT requires O(N log2N) computations
▫ Both compute the same thing
• It should be understood that the FFT is NOT another
transformation but an algorithm to efficiently compute
DFTs.
• Always remember that the signal and its DFT are both
periodic
Improving Resolution in DFT
• When the signal x[n] is periodic of period N, the DFT
values are normalized Fourier series coefficients of x[n]
that only exist for the N harmonic frequencies {2k/N},
as no frequency components exist for any other
frequencies
▫ The number of possible frequencies depend on the
length L chosen to compute its DFT.
▫ Frequencies at which we compute the DFT can be seen
as frequencies around the unit circle in the z-plane
• We can improve the frequency resolution of its DFT by
increasing the number of samples in the signal without
distorting the signal by padding the signal with zeros
▫ Increase samples of the units circle
Zero-Padding Example
Matlab Code
N= 32;
N2= 128;
n=0:N-1;
n2= 0:N2-1;
x= zeros(N,1);
x(1:16)= 1;
X= fftshift(fft(x));
X2= fftshift(fft(x,N2));
subplot(3,1,1)
stem(n,x)
subplot(3,1,2)
stem(n,abs(X))
subplot(3,1,3)
stem(n2,abs(X2))
Original
Signal
32-Point
FFT
128-Point
FFT
No zero-padding
With zero-padding
Phase in DFT
• Output of DFT is complex numbers
• Phase is just as important as magnitude in DFT (in
fact, it is more important!)
• Time shift amounts to linear phase
▫ Phase wrapping occurs for large shifts
• Linear phase can be unwrapped to its correct form
using the “unwrap” function of Matlab
N= 32; x= zeros(N,1); x(11:15)= [0.2 0.5 1 0.5 0.2];
X= fftshift(fft(x));
subplot(4,1,1); stem(1:N,x); title('Signal')
subplot(4,1,2); plot(abs(X)); title('DFT Magnitude')
subplot(4,1,3); plot(angle(X)); title('Computed Phase')
subplot(4,1,4); plot(unwrap(angle(X))); title('Unwrapped Phase')
Phase in DFT: Examples
Less Time Shift
Slower linear phase
More Time Shift
Steeper linear phase
Properties of DFT
• Linearity
x1 n
DFT


X1 k 
x2 n
DFT


X2 k 
ax1 n  bx2 n DFT

 aX1 k   bX2 k 
• Duality
DFT
xn 


DFT
Xn 


Xk 
Nx k N 
Properties of DFT
• Duality: example (n=0:9)
Properties of DFT
• Circular Shift of a Sequence
DFT
xn


X k 
DFT
xn  m N  ; 0  n  N - 1 
 X k e  j 2k / N m
Linear vs. Circular Convolution
• Convolution in continuous and discrete time aperiodic
signals is called linear convolution
• Convolution can be calculated as a multiplication in the
frequency domain
▫ Convolution property of the Fourier transform
• Problem: when using DFT to compute convolution of
aperiodic signals, the periodicity assumption results in
errors in computing the convolution
▫ Signal outside the N points used is not zero: periodic
extension of the N points disturb the computation
▫ What we compute is called circular convolution
Linear vs. Circular Convolution
• Given x[n] and h[n] of lengths M and K, the linear
convolution sum y[n] of length N=(M+K-1) can be found
by following these three steps:
▫ Compute DFTs X[k] and H[k] of length LN for x[n]
and h[n] using zero-padding
▫ Multiply them to get Y[k]= X[k]H[k].
▫ Find the inverse DFT of Y[k] of length L to obtain y[n]
Linear vs. Circular Convolution:
Example
N1= 16;
N2= 16;
N= 32; % N>N1+N2-1
x1= zeros(N1,1);
x2= zeros(N2,1);
x1(1:4)= 1;
x2(:)= 1;
X1= fft(x1,N1);
X2= fft(x2,N1);
X_circ= X1.*X2;
x_circ= ifft(X_circ);
X1= fft(x1,N);
X2= fft(x2,N);
X_lin= X1.*X2;
x_lin= ifft(X_lin);
figure(1); subplot(4,1,1); stem(1:N1,x1)
subplot(4,1,2); stem(1:N2,x2)
subplot(4,1,3); stem(1:N1, abs(x_circ))
title('Circular Convolution');
subplot(4,1,4); stem(1:N,abs(x_lin))
title('Circular Convolution');
Fourier Transformations Chart
Frequency
Signal
Classification
Continuous
Continuous
Discrete
Continuous Fourier
Transform (CFT)
Discrete Fourier Series
(DFS)
Discrete-Time Fourier
Transform (DTFT)
Discrete Fourier
Transform (DFT)
Time
Discrete
Most General Form:
others are special cases
Only one that can be done on a computer
Problem Assignments
• Problems: 10.12, 10.26, 10.27, 10.29
• Partial Solutions available from the student section of
the textbook web site