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Digital Signal Processing – Chapter 10Fourier Analysis of Discrete-Time Signals and Systems
Dr. Ahmed Samir FahmyAssociate ProfessorSystems and Biomedical Eng. Dept.Cairo University
Review 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
Discrete Time Fourier Transform
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 is periodic, with period 2π:
• For convergence:
DTFT
Notes on DTFT
Eigen functions:• Let the input to LTI be: • Using Convolution:
DTFT of h[k] calculated at input frequency
The O/P is similar to I/P (multiplied by a factor)
DTFT vs. Z-Transform
• 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
DTFT vs. Z-Transform
• Consider the noncausal signal , with ▫Determine its DTFT. ▫Use the obtained DTFT to find
• Solution:▫Z-trans:
▫The ROC is: 1st term: 2nd term: Overall ROC:
includes unit circle!
Example
DTFT vs. Z-Transform
• Thus the DTFT is given by
• Also, the DC component is given by:
• Another way to calculate is:
Example
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)
21
0
2
2
( )
k=0, 2N, ...1
0 o.w. 1
kN j nN
n
j k
kjN
X k e
Ne
e
1 1sin( ( 1/ 2))( )
sin( / 2)jwNjw w N
X e ew
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 {2k/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
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))
Matlab Code
OriginalSignal
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')
More Time ShiftLess Time Shift
Steeper linear phaseSlower linear phase
Phase in DFT: Examples
Properties of DFT• Linearity
• Duality
kbXkaXnbxnax
kXnx
kXnx
21DFT
21
2DFT
2
1DFT
1
N
DFT
DFT
kNxnX
kXnx
Properties of DFT• Duality: example (n=0:9)
Properties of DFT• Circular Shift of a Sequence
mNkjDFT
N
DFT
ekXmnx
kXnx/21-Nn0;
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 LN 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+N2-1x1= 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
SignalClassification
Frequency
Continuous Discrete
Time
Continuous Continuous Fourier Transform (CFT)
Discrete Fourier Series (DFS)
Discrete Discrete-Time Fourier Transform
(DTFT)
Discrete Fourier Transform (DFT)
Only one that can be done on a computer
Most General Form: others are special cases
Problem Assignments
• Problems: 10.12, 10.26, 10.27, 10.29• Partial Solutions available from the student
section of the textbook web site