数字信号处理(第四版)第四章ppt

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
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
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Discrete-Time Systems 4.4 Time-domain characteristics of LTI Input-output relationship
Passive and lossless system
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Digital Signal Processing
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Discrete-Time Systems 4.3 Impulse and step responses Unit impulse response, or impulse response
Steps
Zero padding Sliding a window of odd length Median filtering
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Program_4_2.m
Digital Signal Processing
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Discrete-Time Systems 4.2 Classification of DT systems Linear system
Discrete-Time Systems 4.1 Discrete-time system examples (4) Linear Interpolator Linear factor-2 interpolator
Linear factor-3 interpolator
Practical method: Image resize example
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
Example 4.15 Question: For double-side sequences, where is the location of n=0?
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Digital Signal Processing
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Discrete-Time Systems 4.4 Time-domain characteristics of LTI
Finite-dimensional systems
Classification based on impulse response
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Digital Signal Processing
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Discrete-Time Systems Objective of this lecture Function of DT systems
An LTI system is completely characterized by its impulse response. The output of an LTI system for any input sequence is the convolution sum of the input sequence and the impulse response:
Causality condition in terms of impulse response
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Digital Signal Processing
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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
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Discrete-Time Systems 4.1 Discrete-time system examples (4) Linear Interpolator Why: for signal up-sampling or down-sampling, especially for images. Steps
xn
••• 0 (a)
yn
xu n
••• n
••• 0 1 2 3 4 5 6 7 8 9 10 11 12 (b) ••• n
1 2 3
••• 0 1 2 3
4 5 6 7 8 9 10 11 12 (c)
••• n
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Digital Signal Processing
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Digital Signal Processing
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Discrete-Time Systems 4.1 Discrete-time system examples (2) Moving-Average Filter If multiple measurements are available
How can it reduce the noise level?
If measurements cannot be repeated
It is a lowpass filter
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Digital Signal Processing
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Discrete-Time Systems 4.1 Discrete-time system examples (2) Moving-Average Filter Matlab: program_4_1.m (filter, rand)
Digital Signal Processing
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Discrete-Time Systems Outline Discrete-time system examples Classification of DT systems Impulse and step responses Time-domain characteristics of LTI Simple interconnection schemes
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Digital Signal Processing
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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.
Eg. Accumulator form-1 is linear:
Eg. Accumulator form-2 is not linear:
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Digital Signal Processing
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Discrete-Time Systems 4.2 Classification of DT systems Shift invariant system and time shift invariant
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
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Digital Signal Processing
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Discrete-Time Systems 4.4 Time-domain characteristics of LTI
An LTI system is completely characterized by its impulse response
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.
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Digital Signal Processing
ห้องสมุดไป่ตู้
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Discrete-Time Systems 4.2 Classification of DT systems Stable system
A system is stable if and only if for every bounded input, the output is also bounded, called BIBO stable.
Proof:
Properties of convolution
Commutative:
Associative:
Distributive:
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Digital Signal Processing
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Discrete-Time Systems 4.4 Time-domain characteristics of LTI Tabular method of convolution sum computation
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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”?
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Digital Signal Processing
Chapter 04-1-Discretre-Time Systems
Dr. Jimin Liang School of Life Sciences and Technology
Xidian University
jimleung@mail.xidian.edu.cn