Digital Signal Processing Lecture 10 - Discrete Fourier Transform

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1 Digital Signal Processing - Discrete Fourier Transform Electrical Engineering and Computer Science University of Tennessee, Knoxville November 12, 2015

2 Overview

3 Review - 1 Introduction Discrete-time signals and systems - LTI systems Unit sample response h[n]: uniquely characterizes an LTI system Linear constant-coefficient difference equation Frequency response: H(e jω ) Complex exponentials being eigenvalues of an LTI system: y[n] = H(e jω )x[n] Fourier transform z transform The z-transform, X(z) = n= x[n]z n Region of convergence - the z-plane System function, H(z) Properties of the z-transform The significance of zeros The inverse z-transform Relationships between the n, ω, and z domains: Knowing the correspondence between h[n], H(e jω ), and the pole-zero plot

4 Review - 2 Design structures Block diagram vs. Signal flow graph: Knowing how to determine system function, unit sample response, or difference equation from the graphs Different design structures: Knowing pros and cons of each form [Direct form I (zeros first), Direct form II (poles first) - Canonic structure, Transposed direct form II (zeros first), Cascade form, Parallel form, Coupled form] Specific to IIR or FIR: Feedback in IIR (computable vs. noncomputable), Linear phase in FIR Metrics: computational resource and precision Sources of errors: Knowing the concept of pole sensitivity of 2nd-order structures leading to coupled form design, and coefficient quantization examples between direct form vs. cascade form

5 Review - 3 Filter design IIR: CT > DT (impulse invariance vs. bilinear transformation) FIR Knowing the characteristics of the four types of causal linear phase FIR filters Window method - Kaiser window: must use minimum specs; the approximation error is scaled by the size of the jump that produces them Optimal method (Alternation theorem <knowing how to determine the number of alternations>, PM algorithm)

6 Discrete-Time Fourier Transform () Fourier transform representation of x[n] X(e jω ) = x[n] = 1 2π n= π π x[n]e jωn (1) X(e jω )e jωn dω (2) Existence of Fourier transform Absolutely summable (a sufficient condition), n= x[n] < leading to uniform convergence, π lim M π X(ejω ) X M (e jω ) dω = 0 Square summable, n= x[n] 2 <, leading to mean-square convergence, π lim M π X(ejω ) X M (e jω ) 2 dω = 0

7 Discrete Fourier Series () Periodic sequence does not satisfy either absolutely summable or square summable, therefore, it does not have a Fourier representation However, sequences expressed as a sum of complex exponentials can be considered to have an FT representation, i.e., as a train of impulses. X(e jω ) = r= 2πa k δ(ω ω k + 2πr) k Interpret of a periodic signal to be an impulse train in the frequency domain with the impulse values proportional to the coefficients.

8 - cont d X[k] = x[n] = 1 N where W N = e j(2π/n). N 1 n=0 N 1 k=0 x[n]w kn N X[k]W kn N Since any periodic sequence can be represented as a sum of complex exponentials X(e jω ) = k= 2π N X[k]δ(ω 2πk N )

9 Finite-length signal vs. periodic signal x[n] = x[n] p[n] = x[n] δ[n rn] = x[n rn] r= r= x[n] = x[((n)) N ] X[k] = X[((k)) N ]

10 Discrete Fourier Transform () X[k] = N 1 n=0 x[n]w kn N, 0 k N 1 x[n] = 1 N 1 kn N k=0 X[k]WN, 0 n N 1 coefficients are samples of the z-transform at equal-sapce points on the unit circle X(z) = N 1 n=0 x[n]z n, X(k) = X(z) z=w k, k = 0,, N 1 N

11 Relationship between,, and

12 Properties of Shifting properties Duality Convolution property - circular convolution. Think about circular convolution as wrapping the sequence on the surface of two cylinders, one inside another; then convolution is done by rotate the inner cylinder with respect to the outter one. Circular convolution = linear convolution plus aliasing

13 Shifting property

14 Duality property

15 Convolution property

16 Use circular convolution to implement linear convolution

17 Use circular convolution to implement linear convolution - aliasing

18 Use circular convolution to implement linear convolution - zero padding Frequency-domain calculation (a) x 1 [n] X 1 [k], x 2 [n] X 2 [k] (b) X 3 [k] = X 1 [k].x 2 [k] (c) x 3 [n] = x 1 [n]sx 2 [n] = inverse of X 3 [k] (note that I use S to represent circular convolution.) Length of circular convolution x 1 [n] of length L x 2 [n] of length P x 3 [n] = x 1 [n] x 2 [n] of length L + P 1 circular convolution has to be longer than the linear convolution. Time aliasing can be avoided if N L + P 1

19 Example - 2D Convolution

20 Example - 2D Convolution

21 Example - 2D Convolution

22 Discrete Cosine Transform

23 Discrete Cosine Transform

24 Discrete Cosine Transform

Digital Signal Processing Lecture 9 - Design of Digital Filters - FIR

Digital Signal Processing Lecture 9 - Design of Digital Filters - FIR Digital Signal Processing - Design of Digital Filters - FIR Electrical Engineering and Computer Science University of Tennessee, Knoxville November 3, 2015 Overview 1 2 3 4 Roadmap Introduction Discrete-time