Discrete-Time Signals and Systems. Efficient Computation of the DFT: FFT Algorithms. Analog-to-Digital Conversion. Sampling Process.

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1 iscrete-time Signals and Systems Efficient Computation of the FT: FFT Algorithms r. eepa Kundur University of Toronto Reference: Sections 6.1, 6., 6.4, 6.5 of John G. Proakis and imitris G. Manolakis, igital Signal Processing: Principles, Algorithms, and Applications, 4th edition, 007. r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 1 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms / 46 Analog-to-igital Conversion x (t) a A/ converter Sampler Quantizer 6.1 Sampling and Reconstruction of Cts-Time Signals x (n) q Coder Sampling and Reconstruction of Cts-Time Signals To effectively reconstruct an analog from its samples, the sampling frequency 1 must be selected to be large T enough. Sampling in the time-domain: Analog iscrete-time Quantized igital x a (nt ), < n < Sampling: conversion from cts-time to dst-time by taking samples at discrete time instants E.g., uniform sampling: x a (nt ) where T is the sampling period x(t) with interpretation conversion n Return r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 3 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 4 / 46

2 6.1 Sampling and Reconstruction of Cts-Time Signals Time-omain Sampling: frequency-domain perspective The sampling frequency Fs 1 T must be selected to be large enough such that the sampling processing does not cause any loss of spectral information (i.e., no aliasing). Recall CTFT and TFT for aperiodic x a (t) and : x a (t) X a (F ) X a (F )e jft df x a (t)e jft dt X (f ) 1 1 n X (f )e jfn df e jfn 1 1 X (f )e jfn df } } LHS 6.1 Sampling and Reconstruction of Cts-Time Signals x a (nt ) x a ( n ) X a (F )e jf n Fs df } } RHS We can use the following relationship that comes about from the reinterpretation stage. t nt or 1 F T f f T F F f F r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 5 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 6 / Sampling and Reconstruction of Cts-Time Signals 6.1 Sampling and Reconstruction of Cts-Time Signals LHS 1 1 Fs 1 Fs X (f )e jfn df ( F X ( F X ) e j F Fs n df let f F ) e j F Fs n df RHS Fs X a (F )e j F Fs n df + Fs k kfs X a (F )e j Fs [ F Fs n df X a (F k ) e j F kfs n Fs }} X a (F k ) e j F Fs n ] e j F Fs n df let F F + k df r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 7 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 8 / 46

3 6.1 Sampling and Reconstruction of Cts-Time Signals 6.1 Sampling and Reconstruction of Cts-Time Signals Since, 1 Fs ( F X By inspection, or letting f F, X (f ) LHS RHS [ Fs ] X a (F k ) e j F Fs n df ) e j F Fs n df ( ) F X X a (f k ) X a (F k ) X a ((f k) ) The spectrum of the sampled is a scaled periodic repetition of spectrum of the original analog x a (t). The form of the periodic repetition guarantees that the X (f ) is periodic with fundamental period 1 or X (ω) is periodic with fundamental period. See Figure of text. r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 9 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 10 / Sampling and Reconstruction of Cts-Time Signals 6.1 Sampling and Reconstruction of Cts-Time Signals Sampling Theorem Aliasing A bandlimited continuous-time, with highest frequency (bandwidth) B Hz, can be uniquely recovered from its samples provided that the sampling rate B samples per second. Perfect reconstruction is possible via the ideal interpolation formula: See Figure 6.1. of text. x a (t) n x a (nt ) }} sin( (t nt )) T (t nt ) T Sampling and reconstruction of nonbandlimited s results in aliasing. The degree of aliasing/quality of the reconstruction depends on the sampling rate in relation to the decay of the analog spectrum. Example: x a (t) e A t See Figure of text. F X a (F ) A A + (F ), A > 0 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 11 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 1 / 46

4 6. st-time Processing of Cts-Time Signals 6. st-time Processing of Cts-Time Signals Overview Prefilter Analog Prefilter x (t) a A/ st System y(n) /A y (t) a Analog Prefilter x (t) a A/ st System y(n) /A y (t) a system set up when discrete-time processing of continuous-time s is required the application often defines how each block is designed ensures that bandwidth is limited to avoid aliasing or reduce subsequent computational requirements rejects additive noise in higher frequency ranges r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 13 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 14 / st-time Processing of Cts-Time Signals 6. st-time Processing of Cts-Time Signals A/ and /A iscrete-time System esign Analog Prefilter x (t) a A/ st System y(n) /A y (t) a Analog Prefilter x (t) a A/ st System y(n) /A y (t) a ideal sampling and interpolation assumed: y a (t) x(t) tnt x a (nt ) n y(n) sin( T (t nt )) F (t nt ) T F X (F ) 1 T Y a (F ) X a (F k ) TY (F ) F 0 otherwise Q: Is there a discrete-time system such that the overall system above is equivalent to a continuous-time LTI system? A: Yes if x a (t) is bandlimited and > B. See Figure 6.. of text and Figure 6..3 of text. r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 15 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 16 / 46

5 iscrete-time System esign 6. st-time Processing of Cts-Time Signals iscrete-time System esign 6. st-time Processing of Cts-Time Signals See Consider a desired continuous-time LTI system: y a (t) h a (t) x a (t) Y a (F ) H a (F )X a (F ) h a (τ)x a (t τ)dt If x a (t) is bandlimited and > B (no overlap), then Figure 6.. of text X (F ) 1 T X a(f ) for F Fs Recall, from ideal interpolation and assuming dst-time system H(F ): T Y (F ) F Y a (F ) 0 otherwise T H(F )X (F ) F 0 otherwise TH(F ) 1 X T a(f ) F Fs 0 otherwise H(F )Xa (F ) F Fs 0 otherwise r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 17 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 18 / 46 iscrete-time System esign 6. st-time Processing of Cts-Time Signals iscrete-time System esign 6. st-time Processing of Cts-Time Signals The desired equivalent cts-time system is: Y a (F ) H a (F )X a (F ) The actual overall response assuming a dst-time filter H(F ) is: Y a (F ) H(F )Xa (F ) F Fs 0 otherwise Therefore, Naturally, H a (F ) H(F ) F 0 F > Fs H(F ) H a (F ) for F Fs H(F ) H a (F k ) h(n) T h a (nt ) r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 19 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 0 / 46

6 iscrete-time System esign Analog Prefilter x (t) a CT LTI System A/ 6. st-time Processing of Cts-Time Signals st LTI System linear time-varying y(n) /A Under the conditions discussed, the cascade of a linear time-varying system (A/ converter), an LTI system, and a linear time-varying system (/A converter) is equivalent to a continuous-time LTI system. y (t) a 6. st-time Processing of Cts-Time Signals Example: bandlimited differentiator Goal: to design a discrete-time filter such that the overall system (including A/ and /A conversion) is defined by y a (t) dx a(t) dt The frequency response of the overall system is given by: H a (F ) Y a(f ) X a (F ) jf jf F Fc for bandlimited s 0 F > F c r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 1 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms / st-time Processing of Cts-Time Signals Example: bandlimited differentiator Bandlimited Signal Choosing F c, we define the ideal discrete-time differentiator as: and See Figure 6..5 of text. H(F ) H a (F ) jf, H(F ) F H a (F k ) A continuous-time bandpass with bandwidth B and center frequency F c has its frequency content in two frequency bands defined by 0 < F L < F < F H Note: F c F L+F H, B F H F L 1st nd 3rd 4th r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 3 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 4 / 46

7 Uniform First-Order Sampling Integer Band Positioning x a (nt ) F X (F ) 1 T X a (F k ) F H guarantees perfect reconstruction, but wastes bandwidth for bandpass s Consider F H mb where B F H F L Let B results in no aliasing perfect reconstruction is possible with appropriate interpolation stage wasted bandwidth 1st nd 3rd 4th r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 5 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 6 / 46 Integer Band Positioning, B, F H 4B Integer Band Positioning, B, F H 3B 1st nd 3rd 4th 1st nd 3rd r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 7 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 8 / 46

8 Integer Band Positioning Perfect reconstruction via: Arbitrary Band Positioning For B, aliasing is due to overlap of positive spectral band with negative or vice versa. for B. x a (t) n g a (t) sin(bt) Bt x a (nt )g a (t nt ) cos(f c t) r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 9 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 30 / 46 Arbitrary Band Positioning Arbitrary Band Positioning (k 1) F L F L k 1 F H k F H k F H k F L k 1 If obeys the above for integer k 1, aliasing can be avoided. (k 1) F L F H k The k 1 case corresponds to the Nyquist sampling criterion. i.e., Fs F H The case k > 1 corresponds to sampling below Nyquist. i.e., Fs F L /(k 1) < F H for all k > 1. r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 31 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 3 / 46

9 Arbitrary Band Positioning The maximum value of k shows the number of bands that we can fit in the range from 0 and F H. (k 1) F L (k 1) F H B 1 k F H k F H mult. both sides k 1 k kb F H k F H B FH k max B Arbitrary Band Positioning: Minimum Sampling Rate Recall to avoid aliasing, F H k F L k 1 Therefore, the minimum sampling rate to avoid aliasing is given by,min F H k max r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 33 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 34 / 46 Arbitrary Band Positioning Sampling of st-time Signals Therefore to avoid aliasing, the range of acceptable uniform sampling rates is given by F H k F L k 1 where k Z + and 1 k FH B for all n. x d (n) See Figure of text. r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 35 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 36 / 46

10 Sampling of st-time Signals Consider x d (n) for all n can be interpreted as the samples of a continuous-time x a (t) with rate 1 T : X (F ) 1 T X a (F k ) x d (n) can be interpreted as the samples of x a (t) with sampling rate Fs 1, T X d (F ) 1 T X a (F k ) See Figure of text. r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 37 / 46 Sampling of st-time Signals Therefore, X d (F ) 1 1 X k0 ( F k F ) s Assuming X a (F ) 0, F > B, to avoid aliasing for the dst-time sampling, we need: f max B 1 f s B or B or ω max f max ω s r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 38 / 46 Sampling of st-time Signals Interpolation In cts-time sampling x a (nt ), the aperiodic spectrum Xa (F ) is repeated an infinite number of times to create a periodic spectrum covering the infinite frequency range. In dst-time sampling x d (n), the periodic spectrum X (F ) is repeated times covering one period of the periodic frequency domain. Recall, the general interpolation formula is: x a (t) n For dst-time sampling, it is given by: x a (t) m sin( (t nt )) T (t nt ) T x d (m) sin( (t mt )) (t mt ) T T r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 39 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 40 / 46

11 Interpolation Interpolation Using x a (t) m and the fact that x a (nt ) x a (nt ) m m x d (m) sin( (t mt )) (t mt ) m T T x d (m) sin( (nt mt )) (nt mt ) T T x d (m) sin( (n m)) (n m) x d (m)g BL (n m) Therefore, m g BL (n) sin ( n) n x d (m)g BL (n m) r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 41 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 4 / 46 Interpolation Linear Interpolation See Figure 6.5. of text. where g BL (n) sin ( n) n F ω G BL (ω) 0 < ω See Figure of text. x lin (t) x(m 1) + x(m) x(m 1) (t (m 1)T ) T for (m 1)T t mt r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 43 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 44 / 46

12 Linear Interpolation Interpolation Therefore, x lin (n) g lin (n) x(m)g lin (n m) m 1 n n 0 n > where 1 n n g lin (n) 0 n > F G lin (ω) 1 [ ] sin(ω sin( ω) See Figure of text. See Figure 6.5. of text. r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 45 / 46 r. eepa Kundur (University of Toronto) Efficient Computation of the FT: FFT Algorithms 46 / 46

ELEN E4810: Digital Signal Processing Topic 11: Continuous Signals. 1. Sampling and Reconstruction 2. Quantization

ELEN E4810: Digital Signal Processing Topic 11: Continuous Signals 1. Sampling and Reconstruction 2. Quantization 1 1. Sampling & Reconstruction DSP must interact with an analog world: A to D D to A x(t)