Transcript Introduction to Digital Signal Processing

About this Course
Subject:
◦ Digital Signal Processing
◦ EE 541
Textbook
◦ Discrete Time Signal Processing
◦
A. V. Oppenheim and R. W. Schafer, Prentice Hall, 3rd Edition
Reference book
◦ Probability and Random Processes with Applications to Signal Processing
◦
Henry Stark and John W. Woods, Prentice Hall, 3rd Edition
Course website
◦ http://sist.shanghaitech.edu.cn/faculty/luoxl/class/2014Fall_DSP/DSPclass.htm
◦ Syllabus, lecture notes, homework, solutions etc.
About this Course
Grading details:
◦
◦
◦
◦
Homework: (Weekly) 20%
Midterm: 30%
Final: 30%
Project: 20%
Final Score
20%
20%
Homework
Midterm
Final
30%
30%
Project
Matlab
 Powerful software you will like for the rest of your time in
ShanghaiTech SIST
 Ideal for practicing the concepts learnt in this class and doing the final
projects
About the Lecturer
Name: Xiliang Luo (罗喜良)
Research interests:
◦ Wireless communication
◦ Signal processing
◦ Information theory
More information:
◦ http://sist.shanghaitech.edu.cn/faculty/luoxl/
About TA
Name: 裴东
Contact: [email protected]
Office hour: Friday, 6-8pm,
Some survey
 background
 coolest thing you have ever done
 what you want to learn from this course?
Lecture 1:
Introduction to
DSP
XILIANG LUO
2014/9
Signals and Systems
 Signal
1
 something conveying information
0.5
Amplitude
 speech signal
 video signal
 communication signal
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-0.5
 continuous time
-1
chirp
0
0.2
0.4
 discrete time
 digital signal : not only time is discrete, but also is the amplitude!
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Time (secs)
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Discrete Time Signals
 Mathematically, discrete-time signals can be expressed as a sequence of
numbers
𝑥 𝑛 ,𝑛 ∈ 𝑍
 In practice, we obtain a discrete-time signal by sampling a continuoustime signal as:
𝑥 𝑛 = 𝑥𝑎 (𝑛𝑇)
where T is the sampling period and the sampling frequency is defined as
1/T
Speech Signal
Question:
1. What is the sampling frequency?
2. Are we losing anything here by sampling?
Some Basic Sequences
 Unit Sample Sequence
0, 𝑛 ≠ 0
𝛿 𝑛 =
1, 𝑛 = 0
 Unit Step Sequence
𝛿𝑛 =
0, 𝑛 < 0
1, 𝑛 ≥ 0
Some Basic Sequences
 Sinusoidal Sequence
 x 𝑛 = 𝐴 cos(𝜔0 𝑛 + 𝜙)
Question:
1. Is discrete sinusoidal periodic?
2. What is the period?
Question:
Cos(pi/4xn) vs Cos(7pi/4xn), which
One has faster oscillation?
Some Basic Sequences
 Sinusoidal Sequence
Question:
Cos(pi/4xn) vs Cos(7pi/4xn), which
One has faster oscillation?
 x 𝑛 = 𝐴 cos(𝜔0 𝑛 + 𝜙)
1
/4
0.8
7/4
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0.4
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0
-0.2
-0.4
-0.6
-0.8
-1
0
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Discrete-Time Systems
 a transformation or operator mapping discrete time input to discrete time output
𝑦 𝑛 = 𝑇{𝑥[𝑛]}
 Example: ideal delay system
y[n] = x[n-d]
 Example: moving average
y[n] = average{x[n-p],….,x[n+q]}
Memoryless System
 Definition: output at time n depends only on the input at the sample
time n
𝑦 𝑛 =𝑥 𝑛
Question:
Are the following memoryless?
1. y[n] = x[n-d]
2. y[n] = average{x[n-p], …, x[n+q]}
2
Linear System
 Definition: systems satisfying the principle of superposition
𝑇 𝑥1 𝑛 + 𝑥2 𝑛
= 𝑇 𝑥1 [𝑛] + 𝑇{𝑥2 [𝑛]}
𝑇 𝑎𝑥[𝑛] = 𝑎𝑇 𝑥[𝑛]
𝑇 𝑎𝑥1 𝑛 + 𝑏𝑥2 𝑛
= 𝑎𝑇 𝑥1 [𝑛] + 𝑏𝑇{𝑥2 [𝑛]}
Additivity Property
Scaling Property
Superposition Principle
Time-Invariant System
 A.k.a. shift-invariant system: a time shift in the input causes a
corresponding time shift in the output:
𝑇 𝑥[𝑛] = 𝑦[𝑛]
Question:
Are the following time-invariant?
1. y[n] = x[n-d]
2. y[n] = x[Mn]
𝑇 𝑥[𝑛 − 𝑑] = 𝑦[𝑛 − 𝑑]
Causality
 The output of the system at time n depends only on the input
sequence at time values before or at time n;
Is the following system causal?
y[n] = x[n+1] – x[n]
Stability: BIBO Stable
 A system is stable in the Bounded-Input, Bounded-Output (BIBO)
sense if and only if every bounded input sequence produces a bounded
output sequence.
 A sequence is bounded if there exists a fixed positive finite value B
such that:
𝑥 𝑛 ≤𝐵<∞
LTI Systems
 LTI : both Linear and Time-Invariant systems
 convenient representation: completely characterized by its impulse
response
 significant signal-processing applications
 Impulse response
ℎ 𝑛 = 𝑇{𝛿[𝑛]}
 LTI System
𝑥𝑛 =
𝑥 𝑘 𝛿[𝑛 − 𝑘]
𝑘
𝑦 𝑛 =𝑇
𝑥 𝑘 𝛿[𝑛 − 𝑘] =
𝑘
𝑥 𝑘 𝑇{𝛿 𝑛 − 𝑘 ] =
𝑘
𝑥 𝑘 ℎ[𝑛 − 𝑘]
𝑘
LTI System
 LTI system is completely characterized by its impulse response as
follows:
ℎ 𝑛 = 𝑇{𝛿[𝑛]}
𝑦𝑛 =
𝑥 𝑘 ℎ 𝑛−𝑘
𝑘
≜ 𝑥 𝑛 ∗ ℎ[𝑛]
convolution sum
Properties of LTI Systems
 Commutative:
𝑥 𝑛 ∗ ℎ 𝑛 = ℎ 𝑛 ∗ 𝑥[𝑛]
 Distributive:
𝑥 𝑛 ∗ ℎ1 𝑛 + ℎ2 𝑛
 Associative:
= 𝑥 𝑛 ∗ ℎ1 𝑛 + 𝑥 𝑛 ∗ ℎ2 [𝑛]
(𝑥 𝑛 ∗ ℎ1 𝑛 ) ∗ ℎ2 𝑛 = 𝑥 𝑛 ∗ (ℎ1 𝑛 ∗ ℎ2 [𝑛])
Properties of LTI Systems
Equivalent systems:
Properties of LTI Systems
Equivalent systems:
Stability of LTI System
 LTI systems are stable if and only if the impulse response is absolutely
summable:
+∞
|ℎ[𝑘]| < ∞
𝑘=−∞
 sufficient condition
 need to verify bounded input will have also bounded output under this condition
 necessary condition
 need to verify: stable system  the impulse response is absolutely summable
 equivalently: if the impulse response is not absolutely summable, we can prove the system is
not stable!
Stability of LTI System
 Prove: if the impulse response is not absolutely summable, we can
define the following sequence:
ℎ∗ [−𝑛]
, ℎ −𝑛 ≠ 0
𝑥 𝑛 = |ℎ[−𝑛]|
0, ℎ −𝑛 = 0
 x[n] is bounded clearly
 when x[n] is the input to the system, the output can be found to be the
following and not bounded:
𝑦0 =
𝑥 𝑘 ℎ −𝑘 =
ℎ𝑘 2
|ℎ[𝑘]|
Some Convolution Examples
what is the resulting shape?
Matlab cmd: conv()
1
0.9
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10
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1
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Some Convolution Examples
what is the resulting shape?
10
9
700
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600
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500
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3
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400
1
0
0
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100
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0
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Some Convolution Examples
10
9
40
8
what is the freq here?
7
30
6
5
4
20
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2
10
1
0
0
2
4
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0
1
0.8
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-10
0.4
0.2
𝑛𝜋
sin( )
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0
-0.2
-0.4
-0.6
-0.8
-1
0
20
40
60
80
100
120
-20
-30
0
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100
120
Frequency Domain
Representation
 Eigenfunction for LTI Systems
 complex exponential functions are the eigenfunction of all LTI systems
𝑦 𝑛 = 𝑒 𝑗𝜔𝑛 ∗ ℎ 𝑛 =
ℎ 𝑘 𝑒 𝑗𝜔
𝑘
𝐻(𝑒 𝑗𝜔 ) =
ℎ 𝑘 𝑒 −𝑗𝜔𝑘
𝑘
𝑦 𝑛 = 𝐻 𝑒 𝑗𝜔 𝑒 𝑗𝜔𝑛
𝑛−𝑘
= 𝑒 𝑗𝜔𝑛 ×
ℎ 𝑘 𝑒 −𝑗𝜔𝑘
𝑘
Frequency Response of LTE
Systems
 For an LTI system with impulse response h[n], the frequency response
is defined as:
𝐻(𝑒 𝑗𝜔 ) =
ℎ 𝑘 𝑒 −𝑗𝜔𝑘
𝑘
 In terms of magnitude and phase:
𝐻(𝑒 𝑗𝜔 ) = 𝐻 𝑒 𝑗𝜔 𝑒 ∠𝐻(𝑒
𝑗𝜔 )
magnitude response
phase response
Frequency Response of Ideal
Delay
ℎ 𝑛 = 𝛿[𝑛 − 𝑛𝑑 ]
8
-2
6
𝐻 𝑒 𝑗𝜔 =
𝛿 𝑛 − 𝑛𝑑 𝑒 −𝑗𝜔𝑛 = 𝑒 −𝑗𝜔𝑛𝑑
4
phase response
𝑛
2
0
-2
-4
-6
-8
-4
-3
-2
-1
0

1
2
3
4
Frequency Response for a Real
IR
 For real impulse response, we can have:
𝐻 𝑒 −𝑗𝜔 = 𝐻 ∗ (𝑒 𝑗𝜔 )
why?
 Response to a sinusoidal of an LTI with real impulse response
𝑥 𝑛 = Acos(𝜔0 𝑛 + 𝜙 ) =
𝑦𝑛 =
=
𝐴 𝑗(𝜙+𝜔 𝑛) 𝐴 −𝑗(𝜙+𝜔 𝑛)
0
0
𝑒
+ 𝑒
2
2
𝐴
𝐴
𝐻 𝑒 𝑗𝜔𝑜 𝑒 𝑗(𝜙+𝜔0𝑛) + 𝐻(𝑒 −𝑗𝜔0 )𝑒 −𝑗(𝜙+𝜔0𝑛)
2
2
𝐴
𝑗
|𝐻 𝑒 𝑗𝜔𝑜 |𝑒
2
𝜙+𝜔0 𝑛+∠𝐻 𝑒 𝑗𝜔𝑜
+
𝐴
|𝐻 𝑒 𝑗𝜔0 |𝑒 −𝑗(𝜙+𝜔0𝑛+∠𝐻
2
= 𝐻 𝑒 𝑗𝜔𝑜 𝐴 cos(𝜔0 𝑛 + 𝜙 + ∠𝐻(𝑒 𝑗𝜔0 ))
𝑒 𝑗𝜔𝑜 )
Frequency Response Property
 Frequency response is periodic with period 2π
 fundamentally, the following two discrete frequencies are indistinguishable
𝜔, 𝜔 + 2𝜋
 We only need to specify frequency response over an interval of length
2π : [- π, + π];
 In discrete time:
 low frequency means: around 0
 high frequency means: around +/- π
Frequency Response of Typical
Filters
low pass
band-stop
high pass
band-pass
Representation of Sequences
by FT
 Many sequences can be represented by a Fourier integral as follows:
1
𝑥𝑛 =
2𝜋
𝜋
𝑋 𝑒 𝑗𝜔 𝑒 𝑗𝜔𝑛 𝑑𝜔
Synthesis: Inverse Fourier Transform
𝑥[𝑛] 𝑒 −𝑗𝜔𝑛
Analysis: Discrete-Time Fourier Transform
−𝜋
𝑋 𝑒 𝑗𝜔 =
𝑛
 x[n] can be represented as a superposition of infinitesimally small complex
exponentials
 Fourier transform is to determine how much of each frequency component
is used to synthesize the sequence
Prove it!
Convergence of Fourier
Transform
 A sufficient condition: absolutely summable
 it can be shown the DTFT of absolutely summable sequence converge
uniformly to a continuous function
Square Summable
 A sequence is square summable if:
∞
𝑥[𝑛]
2
<∞
𝑛=−∞
 For square summable sequence, we have mean-square convergence:
Ideal Lowpass Filter
DTFT of Complex Exponential
Sequence
 Let a Fourier Transform function be:
 Now, let’s find the synthesized sequence with the above Fourier
Transform:
Symmetry Properties of DTFT
 Conjugate Symmetric Sequence
𝑥𝑒 𝑛 = 𝑥𝑒∗ [−𝑛]
Real  even sequence
 Conjugate Anti-Symmetric Sequence
𝑥𝑜 𝑛 = −𝑥𝑜∗ [−𝑛]
Real  odd sequence
 Any sequence can be expressed as the sum of a CSS and a CASS as
𝑥 𝑛 = 𝑥𝑒 𝑛 + 𝑥𝑜 [𝑛]
How?
Symmetry Properties of DTFT
 DTFT of a conjugate symmetric sequence is conjugate symmetric
 DTFT of a conjugate anti-symmetric sequence is conjugate antisymmetric
 Any real sequence’s DTFT is conjugate symmetric
Fourier Transform Theorems
 Time shifting and frequency shifting theorem
Prove it!
Fourier Transform Theorems
 Time Reversal Theorem
Prove it!
Fourier Transform Theorems
 Differentiation in Frequency Theorem
Prove it!
Fourier Transform Theorems
 Parseval’s Theorem: time-domain energy = freq-domain energy
HW Problem 2.84: Prove a more general format
Fourier Transform Theorems
 Convolution Theorem
Prove it!
Fourier Transform Theorems
 Windowing Theorem
Prove it!
Discrete-Time Random Signals
 Wide-sense stationary random process (assuming real)
𝜙𝑥𝑥 𝑛, 𝑚 = 𝐸 𝑥 𝑛 𝑥[𝑛 + 𝑚] = 𝜙𝑥𝑥 [𝑚]
autocorrelation function
 Consider an LTE system, let x[n] be the input, which is WSS, the output
is denoted as y[n], we can show y[n] is WSS also
Discrete-Time Random Signals
 WSS in, WSS out
Discrete-Time Random Signals
 WSS in, WSS out
Discrete-Time Random Signals
 WSS in, WSS out
Power Spectrum Density
band-pass
White Noise
 Very widely utilized concept in communication and signal processing
 A white noise is a signal for which:
𝜙𝑥𝑥 𝑚 = 𝜎𝑥2 𝛿[𝑚]
Φ𝑥𝑥 𝑒 𝑗𝜔 = 𝜎𝑥2
 From its PSD, we can see the white noise has equal power distribution
over all frequency components
 Often we will encounter the term: AWGN, which stands for: additive
white Gaussian noise
 the underlying random noise is Gaussian distributed
Review
 LTI system
 Frequency Response
 Impulse Response
 Causality
 Stability
 Discrete-Time Fourier Transform
 WSS
 PSD
Homework Problems
2.11 Given LTI frequency response, find the output when input a
sinusoidal sequence …
2.17 Find DTFT of a windowed sequence …
2.22 Period of output given periodic input …
2.40 Determine the periodicity of signals …
2.45 Cascade of LTE systems …
2.51 Check whether system is linear, time-invariant …
2.63 Find alternative system …
2.84 General format of Parseval’s theorem …
Try to use Matlab to plot the sequences and results when required
Next Week
 Z – Transform
 Please read the textbook Chapter 3 in advance!