adsp lec 02
DESCRIPTION
Advanced Digital Signal ProcessingTRANSCRIPT
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Dr. Tahir Zaidi
Advanced Digital Signal Processing
Lecture 2
Signal Representation and Time Domain Analysis
Basic Types of Digital Signals Basic Types of Digital Signals
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Basic Types of Digital Signals
Sine and Exp Using Matlab
% sine generation: A*sin(omega*n+theta)
% exponential generation: A^n
n = 0: 1: 50;
% amplitude
A = 0.87;
% phase
theta = 0.4;
% frequency
omega = 2*pi / 20;
% sin generation
xn1 = A*sin(omega*n+theta);
% exp generation
xn2 = A.^n;
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Basic Operations
Operations in Matlab
xn1 = [1 0 3 2 -1 0 0 0 0 0];
xn2 = [1 3 -1 1 0 0 1 2 0 0];
yn = xn1 + xn2;
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x[n] via impulse functions
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Input: sum of weighted shifted impulses
Time Domain Analysis
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Linear Time-Invariant Systems
Linear
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Linear Time-Invariant Systems Linear Time-Invariant System
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Linear Time-Invariant System
Input: sum of weighted shifted impulses
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Using Linearity and Time-Invariance for the impulses
Sum of wt. Shifted impulses – sum of wt. Shifted impulse responses
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LTI System
Two ways
As the representation of the output as a sum of delayed and scaled impulse responses.
As a computational formula for computing y[n] (“y at time n”) from the entire sequences x and h.
Form x[k]h[n-k] for -∞<k<+∞ for a fixed n
Sum over all k to produce y[n]
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Convolution in the time domain: [ ] [ ] [ ]k
y n x k h n k
y[n] = 2 –3 3 3 –6 0 1 0 0
Example-Convolution of Two Rectangles
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Example..(Continued)
Example-Convolution Of Two Sequences
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Stability
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Causality
Causality & Stability- Example
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Difference Equation
For all computationally realizable LTI systems, the input and output satisfy a difference equation of the form
This leads to the recurrence formula
which can be used to compute the “present” output from the present and M past values of the input and N past values of the output
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Linear Constant-Coefficient Difference(LCCD) Equations
Linear Constant-Coefficient Difference (LCCD) Equations…( Continued)
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Linear Constant-Coefficient Difference (LCCD) Equations….( Continued)
First-Order Example
Consider the difference equation
y[n] =ay[n−1] +x[n]
We can represent this system by the following block diagram:
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Exponential Impulse Response
With initial rest conditions, the difference Equation has impulse response
y[n] =ay[n−1] +x[n]
h[n] =anu[n]
Linear Constant-Coefficient Difference (LCCD) Equations….( Continued)
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Digital Filter
Y = FILTER(B,A,X)
filters the data in vector X with the filter described by
vectors A and B to create the filtered data Y. The filter
is a "Direct Form II Transposed" implementation of the
standard difference equation:
a(1)*y(n) = b(1)*x(n) + b(2)*x(n-1) + ... +
b(nb+1)*x(n-nb) - a(2)*y(n-1) - ... - a(na+1)*y(n-na)
[Y,Zf] = FILTER(B,A,X,Zi)
gives access to initial and final conditions, Zi and Zf, of
the delays.
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LTI summary
Complex Exp Input Signal
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The Frequency Response
Delay and First Difference
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More on the Ideal Delay
Discrete Time Fourier Transform
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Properties
Properties…(cont)
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Symmetric Properties
DTFT Properties:
x[n] X(e j )
x*[n] X*(e j )
x[n] x[n] X(e j ) X(e j )
even even
x[n] x[n] X(e j ) X(e j )
odd odd
x[n] x*[n] X(e j ) X*(e j )
real Hermitian symmetric
Consequences of Hermitian Symmetry
If
then
And
X(e j ) X*(e j )
Re[X(e j )] is even
Im[X(e j )] is odd
X(e j ) is even
X(e j ) is odd
If x[n] is real and even, X(e j ) will be real and even
and if x[n] is real and odd, X(e j ) will be imaginary and odd
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DTFT- Sinusoids
DTFT of Unit Impulse
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Ideal Lowpass Filter
Example
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Magnitude and Angle Form
Magnitude and Angle Plot
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Example
Real and Even ( Zero Phase)
Consider an LTI system with an even unit sample response
DTFT is
e 2 j + 2 e
j + 3 + 2 e j + e
2 j
2 cos( 2 ) + 4 cos( ) + 3
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Real & Even (Zero Phase)
Frequency response is real, so system has “zero” phase shift
This is to be expected since unit sample response is real and even.
Linear Phase
H(z) e2 j + 2e j + 3+3e j + e2 j
e2 j (e2 j + 2e j + 3+2e j + e2 j )
e2 j (2cos(2 )+ 4cos( )+3)
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Useful DTFT Pairs
Convolution Theorem
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Linear Phase… ( cont.)
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Frequency Response of DE
Matlab
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Example
Ideal Filters
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Ideal Filters
HP Digital Filter from LP Design
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BP Digital Filter from HP & LP
Ideal Lowpass Filter
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h[n] of ideal filter
Approximations
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Freq Axis
Inverse System