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ESE 531: Digital Signal Processing
Lec 19: April 6, 2017 Discrete Fourier Transform
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Today
! Adaptive filtering " Blind equalization
! Discrete Fourier Series ! Discrete Fourier Transform (DFT) ! DFT Properties
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Adaptive Filters
! An adaptive filter is an adjustable filter that processes in time " It adapts…
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Adaptive Filter
Update Coefficients
x[n] y[n]
d[n]
e[n]=d[n]-y[n]
+ _
Penn ESE 531 Spring 2017 - Khanna
Adaptive Filter Applications
! System Identification
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Adaptive Filter Applications
! Identification of inverse system
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Adaptive Filter Applications
! Adaptive Interference Cancellation
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Adaptive Filter Applications
! Adaptive Prediction
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Stochastic Gradient Approach
! Most commonly used type of Adaptive Filters ! Define cost function as mean-squared error
" Difference between filter output and desired response
! Based on the method of steepest descent " Move towards the minimum on the error surface to get to
minimum " Requires the gradient of the error surface to be known
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Least-Mean-Square (LMS) Algorithm
! The LMS Algorithm consists of two basic processes " Filtering process
" Calculate the output of FIR filter by convolving input and taps " Calculate estimation error by comparing the output to desired
signal
" Adaptation process " Adjust tap weights based on the estimation error
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Adaptive FIR Filter: LMS
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Adaptive Filter Applications
! Adaptive Interference Cancellation
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Adaptive Interference Cancellation
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Stability of LMS
! The LMS algorithm is convergent in the mean square if and only if the step-size parameter satisfy
! Here λmax is the largest eigenvalue of the correlation matrix
of the input data ! More practical test for stability is
! Larger values for step size
" Increases adaptation rate (faster adaptation) " Increases residual mean-squared error
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max
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λ<µ<
power signal input2
0 <µ<
Discrete Fourier Series
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Reminder: Eigenvalue (DTFT)
! x[n]=ejωn
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y[n]= x[n− k]h[k]k=−∞
∞
∑
= e jω (n−k )h[k]k=−∞
∞
∑
= e jωn h[k]k=−∞
∞
∑ e− jωk
= H (e jω )e jωn
H (e jω ) = h[k]k=−∞
∞
∑ e− jωk
! Describes the change in amplitude and phase of signal at frequency ω
! Frequency response ! Complex value
" Re and Im " Mag and Phase
Discrete Fourier Series
! Definition: " Consider N-periodic signal:
" Frequency-domain also periodic in N:
" “~” indicates periodic signal/spectrum
16 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Discrete Fourier Series
! Define:
! DFS:
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Discrete Fourier Series
! Properties of WN: " WN
0 = WNN = WN
2N = ... = 1 " WN
k+r = WNkWN
r and, WNk+N = WN
k
! Example: WNkn (N=6)
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Discrete Fourier Transform
! By convention, work with one period:
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Same, but different!
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Discrete Fourier Transform
! The DFT
! It is understood that,
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DFS vs. DFT
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Example
22 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Example
23 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Discrete Fourier Series
! Properties of WN: " WN
0 = WNN = WN
2N = ... = 1 " WN
k+r = WNkWN
r and, WNk+N = WN
k
! Example: WNkn (N=6)
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Example
! Q: What if we take N=10? ! A: where is a period-10 seq.
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Example
! Q: What if we take N=10? ! A: where is a period-10 seq.
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Example
! Now, sum from n=0 to 9
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Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Example
! Now, sum from n=0 to 9
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Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
DFT vs. DTFT
! For finite sequences of length N: " The N-point DFT of x[n] is:
" The DTFT of x[n] is:
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DFT vs. DTFT
! The DFT are samples of the DTFT at N equally spaced frequencies
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DFT vs DTFT
! Back to example
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DFT vs DTFT
! Back to example
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DFT vs DTFT
! Back to example
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Use fftshift to center around dc
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
DFT and Inverse DFT
! Use the DFT to compute the inverse DFT. How?
34 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
DFT and Inverse DFT
! Use the DFT to compute the inverse DFT. How?
35 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
DFT and Inverse DFT
! Use the DFT to compute the inverse DFT. How?
36 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
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DFT and Inverse DFT
! Use the DFT to compute the inverse DFT. How?
37 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
DFT and Inverse DFT
! Use the DFT to compute the inverse DFT. How?
38 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
DFT and Inverse DFT
! So
39 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
DFT and Inverse DFT
! So
40 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
DFT and Inverse DFT
! So
! Implement IDFT by:
" Take complex conjugate " Take DFT " Multiply by 1/N " Take complex conjugate
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DFT as Matrix Operator
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DFT as Matrix Operator
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DFT as Matrix Operator
44 N2 complex multiples Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
DFT as Matrix Operator
! Can write compactly as
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Properties of the DFT
! Properties of DFT inherited from DFS ! Linearity
! Circular Time Shift
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Circular Shift
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Properties of DFT
! Circular frequency shift
! Complex Conjugation
! Conjugate Symmetry for Real Signals
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Example: Conjugate Symmetry
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Example: Conjugate Symmetry
50 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Example: Conjugate Symmetry
51 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Example: Conjugate Symmetry
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Example
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Properties of the DFS/DFT
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
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Properties (Continued)
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Circular Convolution
! Circular Convolution:
For two signals of length N
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Circular Convolution
! For x1[n] and x2[n] with length N
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Multiplication
! For x1[n] and x2[n] with length N
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Linear Convolution
! Next.... " Using DFT, circular convolution is easy " But, linear convolution is useful, not circular " So, show how to perform linear convolution with circular
convolution " Use DFT to do linear convolution
59 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Big Ideas
! Adaptive filtering " Use LMS algorithm to update filter coefficients for
applications like system ID, channel equalization, and signal prediction
! Discrete Fourier Transform (DFT) " For finite signals assumed to be zero outside of defined
length " N-point DFT is sampled DTFT at N points " Useful properties allow easier linear convolution
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Admin
! HW 7 out now " Due tonight
! Project posted after class tonight " Work in groups of up to 2
" Can work alone if you want " Use Piazza to find partners " Report your groups to me by 4/11 by email
61 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley