ADSP Lec 02
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Dr. Tahir Zaidi Advanced Digital Signal Processing Spring 2012 Lecture 2 Signal Representation and Time Domain Analysis
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ADSP Lec 02
Transcript of ADSP Lec 02
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Dr. Tahir Zaidi
Advanced Digital Signal Processing
Spring 2012
Lecture 2
Signal Representation and Time
Domain Analysis
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2
Multidimensional Digital Signals
Digital Photography
Digital Video
x1
x2
x1 x2 x (time)
Digital Speech
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Basic Types of Digital SignalsBasic Types of Digital Signals
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2-D Signals
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2-D Unit Impulse Sequence
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2-D Line Impulse Sequence
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2-D Unit Step Sequence
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Basic Types of Digital Signals
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Sine and Exp Using Matlab
n = 0: 1: 50;
% amplitude
A = 0.87;
% phase
theta = 0.4;% frequency
omega = 2*pi / 20;
% sin generationxn1 = A*sin(omega*n+theta);
% exp generation
xn2 = A.^n;
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Basic Operations
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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
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x[n1,n2] via impulse functions
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Time Domain Analysis
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Linear Time-Invariant Systems
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Linear
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Li Ti I i S
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Linear Time-Invariant SystemsLinear Time-Invariant System
Li Ti I i S
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Linear Time-Invariant System
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Input: sum of weighted shifted impulses
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Using Linearity and Time-Invariance for the impulses
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Sum of wt. Shifted impulses – sum of wt. Shifted impulse responses
LTI S t
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LTI System
T
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Two ways
As the representation of the output as asum of delayed and scaled impulseresponses.
As a computational formula forcomputing y[n] (“y at time n”) from theentire 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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2-D Linear Shift Invariant (LSI) System
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2-D Linear Shift Invariant (LSI) System
[ ] [ ] [ ]k h k
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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
E l C l ti f T R t l
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Example-Convolution of Two Rectangles
Example (Continued)
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Example..(Continued)
E l C l ti Of T S
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Example-Convolution Of Two Sequences
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Stability
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Stability
Causality
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Causality
Causality & Stability Example
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Causality & Stability- Example
Properties of Convolution
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Properties of Convolution
Properties of Convolution 2 D
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Properties of Convolution 2-D
Difference Equation
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Difference Equation
For all computationally realizable LTI systems, the
input and output satisfy a difference equation of theform
This leads to the recurrence formula
which can be used to compute the “present” outputfrom the present and M past values of the input andN past values of the output
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Linear Constant-Coefficient Diff Equations LCCD)
First Order Example
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First-Order Example
Consider the difference equationy[n] =ay[n−1] +x[n]
We can represent this system by thefollowing block diagram:
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Linear Constant-Coefficient Diff Equations LCCD)
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Digital Filter
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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.
LTI summary
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LTI summary
Complex Exp Input Signal
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Complex Exp Input Signal
