About this Course Subject: ◦ Digital Signal Processing ◦ EE 541 Textbook ◦ Discrete Time Signal Processing ◦ A. V. Oppenheim and R. W. Schafer, Prentice Hall, 3 rd Edition Reference book ◦ Probability and Random Processes with Applications to Signal Processing ◦ Henry Stark and John W. Woods, Prentice Hall, 3 rd Edition Course website ◦ http:// sist.shanghaitech.edu.cn/faculty/luoxl/class/2014Fall_DSP/DSPclass.htm ◦ Syllabus, lecture notes, homework, solutions etc.
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About this Course Subject: ◦Digital Signal Processing ◦EE 541 Textbook ◦Discrete Time Signal Processing ◦A. V. Oppenheim and R. W. Schafer, Prentice Hall,
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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.
something conveying information speech signal video signal communication signal
continuous time
discrete time digital signal : not only time is discrete, but also is the amplitude!
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plitu
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chirp
Discrete Time Signals Mathematically, discrete-time signals can be expressed as a sequence of numbers
In practice, we obtain a discrete-time signal by sampling a continuous-time signal as:
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
Unit Step Sequence
Some Basic Sequences
Sinusoidal Sequence x
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
x Question:Cos(pi/4xn) vs Cos(7pi/4xn), which One has faster oscillation?
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a transformation or operator mapping discrete time input to discrete time output
Example: ideal delay system y[n] = x[n-d]
Example: moving average y[n] = average{x[n-p],….,x[n+q]}
Discrete-Time Systems
Memoryless System
Definition: output at time n depends only on the input at the sample time n
Question:Are the following memoryless?1. y[n] = x[n-d]2. y[n] = average{x[n-p], …, x[n+q]}
Linear System
Definition: systems satisfying the principle of superposition
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 ] 𝑇 {𝑥 [𝑛−𝑑] }=𝑦 [𝑛−𝑑 ]
Question:Are the following time-invariant?1. y[n] = x[n-d]2. y[n] = x[Mn]
The output of the system at time n depends only on the input sequence at time values before or at time n;
Causality
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
LTI System
𝑥 [𝑛 ]=∑𝑘
❑
𝑥 [𝑘 ]𝛿 [𝑛−𝑘]
h [𝑛 ]=𝑇 {𝛿 [𝑛 ]}
𝑦 [𝑛 ]=𝑇 {∑𝑘
❑
𝑥 [𝑘 ]𝛿 [𝑛−𝑘]}=∑𝑘
❑
𝑥 [𝑘 ]𝑇 {𝛿 [𝑛−𝑘 ] ¿¿=∑𝑘
❑
𝑥 [𝑘 ]h [𝑛−𝑘]
LTI System LTI system is completely characterized by its impulse response as follows:
Stability of LTI System LTI systems are stable if and only if the impulse response is absolutely summable:
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!
∑𝑘=−∞
+∞
¿ h[𝑘]∨¿<∞¿
Stability of LTI System Prove: if the impulse response is not absolutely summable, we can define the following sequence:
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:
𝑥 [𝑛 ]={ h∗ [−𝑛]¿ h[−𝑛]∨¿ , h [−𝑛 ] ≠0¿
0 , h [−𝑛 ]=0
𝑦 [0 ]=∑❑
❑
𝑥 [𝑘 ] h [−𝑘 ]=∑❑
❑ |h [𝑘 ]|2
¿ h[𝑘]∨¿¿
Some Convolution Examples
Matlab cmd: conv()
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what is the resulting shape?
Some Convolution Examples
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Some Convolution Examples
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sin (𝑛𝜋8
)
what is the freq here?
Frequency Domain Representation Eigenfunction for LTI Systems
complex exponential functions are the eigenfunction of all LTI systems
𝑦 [𝑛 ]=𝑒 𝑗 𝜔𝑛∗h [𝑛 ]=∑𝑘
❑
h [𝑘 ]𝑒 𝑗 𝜔 (𝑛−𝑘)=𝑒 𝑗𝜔 𝑛×∑𝑘
❑
h [𝑘 ]𝑒− 𝑗 𝜔𝑘
𝐻 (𝑒 𝑗 𝜔)=∑𝑘
❑
h [𝑘 ]𝑒− 𝑗𝜔 𝑘
𝑦 [𝑛 ]=𝐻 (𝑒 𝑗 𝜔 )𝑒 𝑗𝜔 𝑛
Frequency Response of LTE Systems For an LTI system with impulse response h[n], the frequency response is defined as:
In terms of magnitude and phase:
𝐻 (𝑒 𝑗 𝜔)=∑𝑘
❑
h [𝑘 ]𝑒− 𝑗𝜔 𝑘
𝐻 (𝑒 𝑗 𝜔)=|𝐻 (𝑒 𝑗𝜔 )|𝑒∠𝐻 (𝑒 𝑗 𝜔)
magnitude response
phase response
Frequency Response of Ideal Delay
h [𝑛 ]=𝛿 [𝑛−𝑛𝑑]
𝐻 (𝑒 𝑗 𝜔 )=∑𝑛
❑
𝛿 [𝑛−𝑛𝑑 ]𝑒− 𝑗 𝜔𝑛=𝑒− 𝑗 𝜔𝑛𝑑
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Frequency Response for a Real IR For real impulse response, we can have:
Response to a sinusoidal of an LTI with real impulse response
𝐻 (𝑒− 𝑗 𝜔 )=𝐻∗(𝑒 𝑗 𝜔)why?
𝑥 [𝑛 ]=Acos(𝜔0𝑛+𝜙)=𝐴2𝑒 𝑗 (𝜙+𝜔0𝑛)+
𝐴2𝑒− 𝑗 (𝜙+𝜔0𝑛)
𝑦 [𝑛 ]= 𝐴2𝐻 (𝑒 𝑗𝜔 𝑜)𝑒 𝑗 (𝜙+𝜔0𝑛)+
𝐴2𝐻 (𝑒− 𝑗 𝜔0)𝑒− 𝑗 (𝜙+𝜔0𝑛)
¿𝐴2
∨𝐻 (𝑒 𝑗𝜔 𝑜)∨𝑒 𝑗 (𝜙+𝜔0𝑛+∠𝐻 (𝑒 𝑗 𝜔𝑜 ) )+𝐴2
∨𝐻 (𝑒 𝑗𝜔 0 )∨𝑒− 𝑗 (𝜙+𝜔0𝑛+∠𝐻 (𝑒 𝑗 𝜔𝑜 ))
¿|𝐻 (𝑒 𝑗𝜔 𝑜)|𝐴cos (𝜔0𝑛+𝜙+∠𝐻 (𝑒 𝑗𝜔 0))
Frequency Response Property Frequency response is periodic with period 2π
fundamentally, the following two discrete frequencies are indistinguishable
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
high pass
band-stop
band-pass
Representation of Sequences by FT Many sequences can be represented by a Fourier integral as follows:
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
𝑥 [𝑛 ]= 12𝜋 ∫
−𝜋
𝜋
𝑋 (𝑒 𝑗 𝜔 )𝑒 𝑗𝜔𝑛𝑑𝜔
𝑋 (𝑒 𝑗𝜔 )=∑𝑛
❑
𝑥[𝑛]𝑒− 𝑗𝜔𝑛
Synthesis: Inverse Fourier Transform
Analysis: Discrete-Time Fourier Transform
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:
For square summable sequence, we have mean-square convergence:
∑𝑛=−∞
∞
|𝑥 [𝑛 ]|2<∞
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
Conjugate Anti-Symmetric Sequence
Any sequence can be expressed as the sum of a CSS and a CASS as
𝑥𝑒 [𝑛 ]=𝑥𝑒∗ [−𝑛]
𝑥𝑜 [𝑛 ]=−𝑥𝑜∗ [−𝑛 ]
𝑥 [𝑛 ]=𝑥𝑒 [𝑛 ]+𝑥𝑜[𝑛]
Real even sequence
Real odd sequence
How?
Symmetry Properties of DTFT DTFT of a conjugate symmetric sequence is conjugate symmetric
DTFT of a conjugate anti-symmetric sequence is conjugate anti-symmetric
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)
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