Lec 25: April 28, 2020 Reviewese531/spring2020/handouts/lec25.pdfDiscrete Time Signals & Systems !...
Transcript of Lec 25: April 28, 2020 Reviewese531/spring2020/handouts/lec25.pdfDiscrete Time Signals & Systems !...
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ESE 531: Digital Signal Processing
Lec 25: April 28, 2020 Review
Penn ESE 531 Spring 2020 - Khanna
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Course Content
! Introduction ! Discrete Time Signals & Systems ! Discrete Time Fourier
Transform ! Z-Transform ! Inverse Z-Transform ! Sampling of Continuous Time
Signals ! Frequency Domain of Discrete
Time Series ! Downsampling/Upsampling ! Data Converters, Sigma Delta
Modulation
! Oversampling, Noise Shaping ! Frequency Response of LTI
Systems ! Basic Structures for IIR and FIR
Systems ! Design of IIR and FIR Filters ! Filter Banks ! Adaptive Filters ! Computation of the Discrete
Fourier Transform ! Fast Fourier Transform ! Spectral Analysis ! Wavelet Transform ! Compressive Sampling
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Digital Signal Processing
! Represent signals by a sequence of numbers " Sampling and quantization (or analog-to-digital conversion)
! Perform processing on these numbers with a digital processor " Digital signal processing
! Reconstruct analog signal from processed numbers " Reconstruction or digital-to-analog conversion
• Analog input # analog output • Eg. Digital recording music
• Analog input # digital output • Eg. Touch tone phone dialing, speech to text
• Digital input # analog output • Eg. Text to speech
• Digital input # digital output • Eg. Compression of a file on computer
A/D DSP D/A analog signal
analog signal
digital signal
digital signal
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Discrete Time Signals and Systems
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Signals are Functions
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Discrete Time Systems
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System Properties
! Causality " y[n] only depends on x[m] for m<=n
! Linearity " Scaled sum of arbitrary inputs results in output that is a scaled sum of
corresponding outputs " Ax1[n]+Bx2[n] # Ay1[n]+By2[n]
! Memoryless " y[n] depends only on x[n]
! Time Invariance " Shifted input results in shifted output
" x[n-q] # y[n-q]
! BIBO Stability " A bounded input results in a bounded output (ie. max signal value
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LTI Systems
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! LTI system can be completely characterized by its impulse response
! Then the output for an arbitrary input is a sum of weighted,
delay impulse responses
y[n]= x[n]∗h[n]
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Discrete Time Fourier Transform
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DTFT Definition
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X (e jω ) = x[k]k=−∞
∞
∑ e− jωk
x[n]= 12π
X (e jω−π
π
∫ )e jωndω
X ( f ) = x[k]k=−∞
∞
∑ e− j2π fk
x[n]= X ( f−0.5
0.5
∫ )e j2π fndfAlternate
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Example: Window DTFT
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W (e jω ) = w[k]k=−∞
∞
∑ e− jωk
= e− jωkk=−N
N
∑
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Example: Window DTFT
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W (e jω ) =sin (N +1 2)ω( )sin ω 2( )
=1 why?
Also, Σx[n]
Plot for N=2
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LTI System Frequency Response
! Fourier Transform of impulse response
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H (e jω ) = h[k]k=−∞
∞
∑ e− jωk
LTI System x[n]=ejωn y[n]=H(ejωn)ejωn
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z-Transform
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! The z-transform generalizes the Discrete-Time Fourier Transform (DTFT) for analyzing infinite-length signals and systems
! Very useful for designing and analyzing signal processing systems
! Properties are very similar to the DTFT with a few caveats
H (z) = h[n]n=−∞
∞
∑ z−n
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Region of Convergence (ROC)
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Inverse z-Transform
! Ways to avoid it: " Inspection (known transforms) " Properties of the z-transform " Partial fraction expansion
" Power series expansion
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X (z) =b0a0
(1− ck z−1)
k=1
M
∏
(1− dk z−1)
k=1
N
∏=
Ak1− dk z
−1k=1
N
∑
X (z) = x[n]z−nn=−∞
∞
∑
=!+ x[−2]z2 + x[−1]z + x[0]+ x[1]z−1 + x[2]z−2 +!
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Difference Equation to z-Transform
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ak y[n− k]k=0
N
∑ = bmx[n−m]m=0
M
∑
! Difference equations of this form behave as causal LTI systems " when the input is zero prior to n=0 " Initial rest equations are imposed prior to the time when input
becomes nonzero " i.e y[-N]=y[-N+1]=…=y[-1]=0
H (z) =bk( ) z−k
m=0
M
∑
ak( ) z−kk=0
N
∑
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Sampling and Reconstruction
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DSP System
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Ideal Sampling Model
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! Ideal continuous-to-discrete time (C/D) converter " T is the sampling period " fs=1/T is the sampling frequency " Ωs=2π/T
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Ideal Sampling Model
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Frequency Domain Analysis
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Xs (ejΩ)
Ωs
2⋅T = π
ω =ΩT
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Frequency Domain Analysis
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Xs (ejΩ)
Ωs
2⋅T = π
ω =ΩT
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Aliasing Example
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Reconstruction of Bandlimited Signals
! Nyquist Sampling Theorem: Suppose xc(t) is bandlimited. I.e.
! If Ωs≥2ΩN, then xc(t) can be uniquely determined from its samples x[n]=xc(nT)
! Bandlimitedness is the key to uniqueness
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Mulitiple signals go through the samples, but only one is
bandlimited within our sampling band
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Reconstruction in Frequency Domain
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Reconstruction in Time Domain
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*
= The sum of “sincs”
gives xr(t) # unique signal that is
bandlimited by sampling bandwidth
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Anti-Aliasing Filter
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1
-ΩN ΩN
XC ( jΩ)X LP ( jΩ)
ΩN
ΩS/2
XS ( jΩ)1/T
1
-ΩN ΩN
XC ( jΩ)
ΩS/2
ΩN
ΩS/2
XS ( jΩ)1/T
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Rate Re-Sampling
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Downsampling
! Definition: Reducing the sampling rate by an integer number
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Example: M=2
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4π 2π
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Example: M=3
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6π 4π 2π
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Example: M=3 w/ Anti-aliasing
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Upsampling
! Definition: Increasing the sampling rate by an integer number
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x[n]= xc (nT )xi[n]= xc (nT ')
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Upsampling
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xi[n]
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Example
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Non-integer Sampling
! T’=TM/L " Upsample by L, then downsample by M
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interpolator decimator
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Interchanging Operations
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Upsampling -expanding in time -compressing in frequency
Downsampling -compressing in time -expanding in frequency
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Interchanging Operations - Summary
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Filter and expander Expander and expanded filter*
Compressor and filter Expanded filter* and compressor
*Expanded filter = expanded impulse response, compressed freq response
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Polyphase Implementation of Decimator
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interpolator decimator
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Polyphase Implementation of Interpolation
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interpolator decimator
E0(z)
E0(z)
E0(z)
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Multi-Rate Filter Banks
! Use filter banks to operate on a signal differently in different frequency bands " To save computation, reduce the rate after filtering
! h0[n] is low-pass, h1[n] is high-pass " Often h1[n]=ejπnh0[n] $ shift freq resp by π
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Multi-Rate Filter Banks
! Assume h0, h1 are ideal low/high pass with ωC=π/2
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Have to be careful with
order!
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Perfect Reconstruction non-Ideal Filters
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Quadrature Mirror Filters
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Quadrature mirror filters
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Frequency Response of Systems
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Frequency Response of LTI System
! We can define a magnitude response
! And a phase response
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Y e jω( ) = H e jω( ) X e jω( )
Y e jω( ) = H e jω( ) X e jω( )
∠Y e jω( ) =∠H e jω( )+∠X e jω( )
Penn ESE 531 Spring 2020 – Khanna Adapted from M. Lustig, EECS Berkeley
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Group Delay
! General phase response at a given frequency can be characterized with group delay, which is related to phase
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ωω1 ω2
- slope Penn ESE 531 Spring 2020 – Khanna Adapted from M. Lustig, EECS Berkeley
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LTI System
! Transfer function is not unique without ROC " If diff. eq represents LTI and causal system, ROC is
region outside all singularities " If diff. eq represents LTI and stable system, ROC
includes unit circle in z-plane 49 Penn ESE 531 Spring 2020 - Khanna
Stable and causal if all poles inside
unit circle
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General All-Pass Filter
! dk=real pole, ek=complex poles paired w/ conjugate, ek
*
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Minimum-Phase Systems
! Definition: A stable and causal system H(z) (i.e. poles inside unit circle) whose inverse 1/H(z) is also stable and causal (i.e. zeros inside unit circle) " All poles and zeros inside unit circle
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H(z) 1/H(z)
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Min-Phase Decomposition Purpose
! Have some distortion that we want to compensate for:
! If Hd(z) is min phase, easy: " Hc(z)=1/Hd(z) $ also stable and causal
! Else, decompose Hd(z)=Hd,min(z) Hd,ap(z) " Hc(z)=1/Hd,min(z) #Hd(z)Hc(z)=Hd,ap(z)
" Compensate for magnitude distortion
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Generalized Linear Phase
! An LTI system has generalized linear phase if frequency response can be expressed as:
! Where A(ω) is a real function.
! What is the group delay?
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FIR GLP: Type I and II
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FIR GLP: Type III and IV
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Zeros of GLP System
! FIR GLP System Function
! If zero is on unit circle (r=1)
! If zero is real and not on unit circle (θ=0)
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FIR Filter Design
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FIR Design by Windowing
! Given desired frequency response, Hd(ejω) , find an impulse response
! Obtain the Mth order causal FIR filter by truncating/windowing it
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FIR Design by Windowing
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Tapered Windows
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Tradeoff – Ripple vs. Transition Width
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Optimality
! Least Squares:
! Variation: Weighted Least Squares:
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Min-Max Ripple Design
! Recall, is symmetric and real ! Given ωp, ωs, M, find δ,
! Formulation is a linear program with solution δ, ! A well studied class of problems
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Discrete Fourier Transform
DFT
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Discrete Fourier Transform
! The DFT
! It is understood that,
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X [k]= Wnk10n=0
5∑ = e
− jπ2ksin 3π
5k
⎛
⎝⎜
⎞
⎠⎟
sin π10k
⎛
⎝⎜
⎞
⎠⎟
DFT vs DTFT
! Back to example
66 Penn ESE 531 Spring 2020 – Khanna Adapted from M. Lustig, EECS Berkeley
“10-point” DFT
Use fftshift to center around dc
“6-point” DFT
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Circular Convolution
! For x1[n] and x2[n] with length N
" Very useful!! (for linear convolutions with DFT)
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Linear Convolution via Circular Convolution
! Zero-pad x[n] by P-1 zeros
! Zero-pad h[n] by L-1 zeros
! Now, both sequences are length M=L+P-1
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Block Convolution
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Example of Overlap-Add
70
y
y
y
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L=11
L+P-1=16
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Example of Overlap-Save
71 Penn ESE 531 Spring 2020 – Khanna Adapted from M. Lustig, EECS Berkeley
L+P-1=16 P-1=5 Overlap samples
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Circular Conv. as Linear Conv. w/ Aliasing
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! Therefore
! The N-point circular convolution is the sum of linear convolutions shifted in time by N
x3p[n]= x1[n]⊗ x2[n]N
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Example:
! Let
! What does the L-point circular convolution look like?
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Linear convolution
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Example:
! The L-shifted linear convolutions
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Fast Fourier Transform
FFT
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Fast Fourier Transform
! Enable computation of an N-point DFT (or DFT-1) with the order of just N· log2 N complex multiplications.
! Most FFT algorithms decompose the computation of a DFT into successively smaller DFT computations. " Decimation-in-time algorithms " Decimation-in-frequency
! Historically, power-of-2 DFTs had highest efficiency ! Modern computing has led to non-power-of-2 FFTs with
high efficiency ! Sparsity leads to reduce computation on order K· logN
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Decimation-in-Time FFT
77 Penn ESE 531 Spring 2020 – Khanna Adapted from M. Lustig, EECS Berkeley
• 3=log2(N)=log2(8) stages • 4=N/2=8/2 multiplications in each stage
• 1st stage has trivial multiplication
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Decimation-in-Frequency FFT
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Admin
! Final Project due – Today! " TA advice – “The report takes time. Leave time for it.”
! Last day of TA office hours – Apr 28th
" Piazza still available " Review session for exam TBD
" Poll in Piazza coming soon
! Last day of Tania office hours – May 1st ! Final Exam – May 7th
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Final Exam Admin
! Final Exam – 5/7 (3pm-5pm) " In Canvas
" Will have a 2 hr window to complete within a 14 hr time block (6am-8pm EDT)
" Open course notes and textbook, but cannot communicate with each other about the exam
" Students will have randomized and different questions " Reminder, it is not in your best interest to share the exam
" Old exams posted on old course websites " Covers Lec 1- 20
" Does not include lec 12 (data converters and noise shaping) or IIR Filters
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