Fast Fourier Transform Fourier Series & Transform Summary - PowerPoint PPT Presentation

Fast Fourier Transform Fourier Series & Transform Summary • Discrete-time windowing � X [ k ] e jk Ω o n x [ n ] = • Discrete Fourier Transform k = <N> • Relationship to DTFT X [ k ] = 1 � x [ n ]e − jk Ω o n N • Relationship to DTFS n = <N> • Zero padding x [ n ] = 1 � X (e jω ) e j Ω n dΩ 2 π 2 π ∞ � X (e jω ) = x [ n ] e − j Ω n n = −∞ • What are the similarities and differences between the DTFS & DTFT? J. McNames Portland State University ECE 223 FFT Ver. 1.03 1 J. McNames Portland State University ECE 223 FFT Ver. 1.03 2 Periodic Signals Example 1: Relationship to Fourier Series Suppose that we have a periodic signal x p [ n ] with fundamental period ∞ FT � � X [ k ] e jk Ω o n N . Define the truncated signal x [ n ] as follows. x [ n ] = ⇐ ⇒ 2 π X [ k ] δ (Ω − k Ω o ) k = <N> k = −∞ � x p [ n ] n 0 + 1 ≤ n ≤ n 0 + N x [ n ] = • Recall that DT periodic signals can be represented by the DTFT 0 otherwise • Requires the use of impulses (why?) Determine how the Fourier transform of x [ n ] is related to the • This makes the DTFT more general than the DTFS discrete-time Fourier series coefficients of x p [ n ] . Recall that X p [ k ] = 1 � x p[ n ]e − jk (2 π/N ) n N n = <N> J. McNames Portland State University ECE 223 FFT Ver. 1.03 3 J. McNames Portland State University ECE 223 FFT Ver. 1.03 4

Example 1: Workspace DFT Estimate of DTFT � w [ n ] x [ n ] 0 ≤ n ≤ N − 1 x w [ n ] = x [ n ] · w [ n ] = 0 Otherwise X w (e jω ) = 1 2 π X (e jω ) ⊛ W (e jω ) • Recall that windowing in the time domain is equivalent to filtering (convolution) in the frequency domain • We found earlier that a windowed signal x w [ n ] = x [ n ] · w [ n ] could be thought of as one period of a periodic signal x p [ n ] • This enables us to calculate the DTFT at discrete-frequencies using the discrete-time Fourier series analysis equation! • Advantages – DTFS consists of a finite sum - we can calculate it – DTFS can be calculated very efficiently using the Fast Fourier Transform (FFT) J. McNames Portland State University ECE 223 FFT Ver. 1.03 5 J. McNames Portland State University ECE 223 FFT Ver. 1.03 6 FFT Estimate of DTFT Derived DFT, FFT, and DTFS X [ k ] = 1 + ∞ � x [ n ]e − jk Ω o n DTFS X w (e jω ) � x w [ n ]e − j Ω n = N n = <N> n = −∞ N − 1 N − 1 � x w [ n ]e − jk 2 π N n DFT/FFT X w [ k ] = � x w [ n ] e − j Ω n = n =0 n =0 • Note abuse of notation, X [ k ] N − 1 x w [ n ]e − jk 2 π X w (e jω ) � N n � = • The DFT is a transform � Ω= k 2 π N n =0 • The FFT is a fast algorithm to calculate the DFT = DFT { x w [ n ] } • If x [ n ] is a periodic signal with fundamental period N , then the • Calculation of the DTFT of a finite-duration signal at discrete DFT is the same as the scaled DTFS frequencies is called the Discrete Fourier Transform (DFT) • The DFT can be applied to non-periodic signals • The FFT is just a fast algorithm for calculating the DFT – For non-periodic signals this is modelled with windowing • The DTFS cannot J. McNames Portland State University ECE 223 FFT Ver. 1.03 7 J. McNames Portland State University ECE 223 FFT Ver. 1.03 8

Example 2: FFT Estimate of DTFT Example 2: Workspace Solve for the Fourier transform of ⎧ − 1 1 ≤ n ≤ 4 ⎪ ⎨ x [ n ] = +1 13 ≤ n ≤ 16 ⎪ 0 Otherwise ⎩ J. McNames Portland State University ECE 223 FFT Ver. 1.03 9 J. McNames Portland State University ECE 223 FFT Ver. 1.03 10 Example 2: Workspace Example 2: Signal 1 0.8 0.6 0.4 0.2 x[n] 0 −0.2 −0.4 −0.6 −0.8 −1 0 2 4 6 8 10 12 14 16 Time J. McNames Portland State University ECE 223 FFT Ver. 1.03 11 J. McNames Portland State University ECE 223 FFT Ver. 1.03 12

Example 2: DTFT Estimate Zero Padding X w (e jω ) � N = DFT { x w [ n ] } 5 � Ω= k 2 π Real X(e j ω ) 0 • The FFT has two apparent disadvantages – It requires that N be an integer power of 2: N = 2 ℓ −5 True DTFT DFT Estimate – It only generates estimates at N frequencies equally spaced −10 between 0 and 2 π rad/sample 0 0.5 1 1.5 2 2.5 3 • Both of these problems can be circumvented by zero-padding 10 • Recall that x w [ n ] = x [ n ] · w [ n ] Imag X(e j ω ) 5 • We can choose N to be larger than the length of our window w [ n ] 0 −5 0 0.5 1 1.5 2 2.5 3 Frequency (rads per sample) J. McNames Portland State University ECE 223 FFT Ver. 1.03 13 J. McNames Portland State University ECE 223 FFT Ver. 1.03 14 Zero Padding Derived Example 3: FFT Estimate of DTFT Repeat the previous example but use zero-padding so that the estimate Suppose we add M − N zeros to the finite-length signal x [ n ] such is evaluated at no less than 1000 frequencies between 0 and π . that M is an integer power of 2 and M ≥ N . Then the zero-padded signal, x z [ n ] , has a length M . The frequency resolution then improves to 2 π M rad/sample, rather than 2 π N rad/sample. N − 1 x w [ n ]e − j ( k 2 π N ) n � DFT { x w [ n ] } = n =0 M − 1 x z [ n ]e − j ( k 2 π M ) n � DFT { x z [ n ] } = n =0 N − 1 x w [ n ]e − j ( k 2 π M ) n � = n =0 X w (e jω ) � = DFT { x z [ n ] } � Ω= k 2 π M J. McNames Portland State University ECE 223 FFT Ver. 1.03 15 J. McNames Portland State University ECE 223 FFT Ver. 1.03 16

Example 3: DTFT Estimate with Padding Example 3: MATLAB Code %function [] = FFTEstimate(); 5 close all; n = 0:17; Real X(e j ω ) x = -1*(n>=1 & n<=4) + 1*(n>=13 & n<=16); 0 %============================================================================== % Plot the signal %============================================================================== −5 figure True DTFT FigureSet(1,4.5,2.8); DFT Estimate plot([min(n) max(n)],[0 0],’k:’); hold on; −10 h = stem(n,x,’b’); 0 0.5 1 1.5 2 2.5 3 set(h(1),’MarkerFaceColor’,’b’); set(h(1),’MarkerSize’,4); hold off; 10 ylabel(’x[n]’); xlabel(’Time’); xlim([min(n) max(n)]); Imag X(e j ω ) ylim([-1.05 1.05]); 5 box off; AxisSet(8); print -depsc FFTESignal; 0 %============================================================================== % Plot the True Transform \& Estimate %============================================================================== n = 1:16; −5 x = -1*(n>=1 & n<=4) + 1*(n>=13 & n<=16); 0 0.5 1 1.5 2 2.5 3 N = length(x); k = 0:N-1; Frequency (rads per sample) we = (0:N-1)*(2*pi/N); % Frequency of estimates Xe = exp(-j*k*(2*pi/N)*1).*fft(x); J. McNames Portland State University ECE 223 FFT Ver. 1.03 17 J. McNames Portland State University ECE 223 FFT Ver. 1.03 18 w = 0.0001:(2*pi)/1000:2*pi; figure X = (-exp(-j*w*1) + exp(-j*w*5) + exp(-j*w*13) - exp(-j*w*17))./(1-exp(-j*w)); % True spectrum FigureSet(1,’LTX’); subplot(2,1,1); figure h = plot(w,real(X),’r’,we,real(Xe),’k’); FigureSet(1,’LTX’); set(h(1),’LineWidth’,2.0); subplot(2,1,1); set(h(2),’LineWidth’,1.0); h = plot(w,real(X),’r’,we,real(Xe),’k’); ylabel(’Real X(e^{j\omega})’); set(h(2),’Marker’,’o’); xlim([0 pi]); set(h(2),’MarkerFaceColor’,’k’); box off; set(h(2),’MarkerSize’,3); AxisLines; ylabel(’Real X(e^{j\omega})’); legend(h(1:2),’True DTFT’,’DFT Estimate’,4); xlim([0 pi]); subplot(2,1,2); box off; h = plot(w,imag(X),’r’,we,imag(Xe),’k’); AxisLines; legend(h(1:2),’True DTFT’,’DFT Estimate’,4); set(h(1),’LineWidth’,2.0); subplot(2,1,2); set(h(2),’LineWidth’,1.0); h = plot(w,imag(X),’r’,we,imag(Xe),’k’); ylabel(’Imag X(e^{j\omega})’); set(h(2),’Marker’,’o’); xlim([0 pi]); set(h(2),’MarkerFaceColor’,’k’); box off; set(h(2),’MarkerSize’,3); AxisLines; ylabel(’Imag X(e^{j\omega})’); xlabel(’Frequency (rads per sample)’); xlim([0 pi]); AxisSet(8); box off; print -depsc FFTEstimatePadded; AxisLines; xlabel(’Frequency (rads per sample)’); AxisSet(8); print -depsc FFTEstimate; %============================================================================== % Plot the True Transform \& Estimate %============================================================================== n = 1:16; x = -1*(n>=1 & n<=4) + 1*(n>=13 & n<=16); N = 2^(ceil(log2(2000))); k = 0:N-1; we = (0:N-1)*(2*pi/N); % Frequency of estimates Xe = exp(-j*k*(2*pi/N)*1).*fft(x,N); J. McNames Portland State University ECE 223 FFT Ver. 1.03 19 J. McNames Portland State University ECE 223 FFT Ver. 1.03 20

Some Key Points on Windowing & the FFT • In practice, most DT signals are truncated to a finite duration prior to processing • Mathematically, this is equivalent to multiplying an signal with infinite duration by another signal with finite duration: x [ n ] · w [ n ] • This process is called windowing • Windowing in time is equivalent to convolution in frequency: FT 1 � X (e jω ) ⊛ W (e jω ) � x [ n ] · w [ n ] ⇐ ⇒ 2 π • This causes a blurring of the estimated DTFT • The FFT is a very efficient means of calculating the DTFT of a finite duration DT signal at discrete frequencies • Zero padding can be used to obtain arbitrary resolution • The FFT can also be used to efficiently perform convolution in time (details were not discussed in class) J. McNames Portland State University ECE 223 FFT Ver. 1.03 21