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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Systems 4.1 Discrete-time system examples (2) Moving-Average Filter If multiple measurements are available
Causal system
The n_0 output sample y[n_0] depends only on input samples x[n] for n<=n_0, and do not depend on input samples for n>n_0 For a causal system, changes in the output samples do not precede changes in the input samples Interpolator is not a causal system
Passive and lossless system
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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Systems 4.3 Impulse and step responses Unit impulse response, or impulse response
Eg. Accumulator form-1 is linear:
Eg. Accumulator form-2 is not linear:
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Digital Signal iang
Discrete-Time Systems 4.2 Classification of DT systems Shift invariant system and time shift invariant
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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Systems 4.1 Discrete-time system examples (1) Accumulator Form 1: Form 2: Form 3:
Corresponding with the integral for analog signal
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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Systems 4.4 Time-domain characteristics of LTI
An LTI system is completely characterized by its impulse response
Example 4.15 Question: For double-side sequences, where is the location of n=0?
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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Systems 4.4 Time-domain characteristics of LTI
© 2013 Jimin Liang
Discrete-Time Systems 4.1 Discrete-time system examples (3) Exponentially Weighted Running Average Filter Why: Place more emphasis on recent data samples and less emphasis on samples that are further away. Why call it “exponentially”?
Causality condition in terms of impulse response
8 7 6 5 5
Amplitude
8 d[n] s[n] x[n] 7 6 s[n] y[n]
4 3 2 1 0 -1
Amplitude
4 3 2 1 0
0
5
10
15
20 25 30 Time index n
35
40
45
50
0
5
10
15
20 25 30 Time index n
35
40
45
50
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Digital Signal Processing
The response of a system to a unit sample sequence.
Unit step response, or unit response Eg. 4.9, 4.10
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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Systems 4.4 Time-domain characteristics of LTI Input-output relationship
Proof:
Properties of convolution
Commutative:
Associative:
Distributive:
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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Systems 4.4 Time-domain characteristics of LTI Tabular method of convolution sum computation
© 2013 Jimin Liang
Digital Signal Processing
Chapter 04-1-Discretre-Time Systems
Dr. Jimin Liang School of Life Sciences and Technology
Xidian University
jimleung@
Steps
Zero padding Sliding a window of odd length Median filtering
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Program_4_2.m
Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Systems 4.2 Classification of DT systems Linear system
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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Systems 4.1 Discrete-time system examples (4) Linear Interpolator Why: for signal up-sampling or down-sampling, especially for images. Steps
Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Systems Outline Discrete-time system examples Classification of DT systems Impulse and step responses Time-domain characteristics of LTI Simple interconnection schemes
How can it reduce the noise level?
If measurements cannot be repeated
It is a lowpass filter
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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Systems 4.1 Discrete-time system examples (2) Moving-Average Filter Matlab: program_4_1.m (filter, rand)
An LTI system is completely characterized by its impulse response
Stability condition in terms of impulse response
An LTI system is BIBO stable if and only if its impulse response sequence is absolutely summable.
Process a given sequence, called the input system, to generate another sequence, called the output sequence, with more desirable properties or to extract certain information about the input signal. DT system is usually also called the digital filter
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Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Systems 4.1 Discrete-time system examples
(5) Median filter
Why: remove additive impulse noise Definition: The median of a set of (2K+1) numbers is the number such that K number form the set have values greater than this number, while other K numbers have values smaller.