J. McNames Portland State University ECE 223 DT Fourier Series Ver

Transcription

1 Overview of DT Fourier Series Topics Orthogonality of DT exponential harmonics DT Fourier Series as a Design Task Picking the frequencies Picking the range Finding the coefficients Example J. McNames Portland State University ECE 223 DT Fourier Series Ver

2 Motivation h[n] x[n] y[n] x[n] H(e jω ) y[n] e jωn H(e jωn )e jωn X[k] e jωkn X[k] H(e jωkn )e jω kn k k H(e jω ) = F {h[n]} = h[n] e jωn n= For now, we restrict out attention to DT periodic signals: x[n + N] = x[n] Would like to represent x[n] as a sum of complex sinusoids Why? Gives us insight and simplifies computation J. McNames Portland State University ECE 223 DT Fourier Series Ver

3 LTI Systems: Key Property Linearity is a key property of LTI systems (such as circuits) A system is linear if for any two signals, say x 1 [n] and x 2 [n] x 1 [n] y 1 [n] x 2 [n] y 2 [n] and any linear combination of these two signals, say x[n] α 1 x 1 [n] + α 2 x 2 [n] y[n] α 1 y 1 [n] + α 2 y 2 [n] The immediate generalization that follows is if x k [n] y k [n] for all k then X[k]x k [n] k k X[k]y k [n] J. McNames Portland State University ECE 223 DT Fourier Series Ver

4 Periodic Sum Notation Suppose x[n] is a discrete-time (DT) periodic signal with fundamental period N. Then the sum of x[n] over a fundamental period is the same no matter what time the sum begins s = N 1 x[n] = N x[n] = N 1+n 0 x[n] n=0 n=1 n=n 0 We will denote the sum over any fundamental period of the signal generically as s = x[n] J. McNames Portland State University ECE 223 DT Fourier Series Ver

5 Jointly Periodic Signals Suppose two DT periodic signals x 1 [n] and x 2 [n] have a common fundamental period N, defined as the smallest N such that both x 1 [n] = x 1 [n + N] and x 2 [n] = x 2 [n + N] I will refer to these signals as jointly periodic The product p[n] x 1 [n]x 2 [n] is then also periodic with fundamental period N: p[n + N] = x 1 [n + N]x 2 [n + N] = x 1 [n]x 2 [n] = p[n] J. McNames Portland State University ECE 223 DT Fourier Series Ver

6 Inner Products and Orthogonality Let us define an inner product of two DT jointly periodic signals as < x 1 [n], x 2 [n] > x 1 [n]x 2[n] where denotes complex conjugate. The norm of a signal is the inner product of the signal with itself x[n] 2 < x[n], x[n] >= x[n]x [n] = x[n] 2 Two signals are said to be orthogonal if and only if their inner product is zero x 1 [n]x 2[n] = 0 J. McNames Portland State University ECE 223 DT Fourier Series Ver

7 Signal Decompositions Suppose we can represent a DT periodic signal x[n] as a linear combination of basis functions x[n] = K 1 k=0 This may be useful for several reasons X[k]x k [n] If we know how an LTI system responds to each x k [n], say x k [n] y k [n], we immediately know: x[n] y[n] = K 1 k=0 X[k]y k [n] The set of coefficients X[k] is a complete and, possibly, compact representation of the signal (signal compression) If the basis functions x k [n] are meaningful, the coefficients X[k] indicate the degree to which each contributes to x[n] J. McNames Portland State University ECE 223 DT Fourier Series Ver

8 Strategy for Finding the Coefficients x[n] = K 1 k=0 X[k]x k [n] Problem: for a given periodic signal x[n] and known set of basis functions {x k [n]}, how do we determine the coefficients X[k]? This becomes a relatively easy problem if the basis functions x k [n] are orthogonal to one another just take the inner product J. McNames Portland State University ECE 223 DT Fourier Series Ver

9 Solving for the Coefficients with Inner Products For the sake of simplicity, let us assume < x k [n], x k [n] >= x k [n] 2 = c for all k. Then < x[n], x l [n] > = x[n]x l [n] = = = K 1 k=0 K 1 k=0 = cx[l] X[k] ( K 1 k=0 X[k]c δ(k l) X[l] = 1 c < x[n], x l[n] > X[k]x k [n] ) x k [n]x l[n] x l[n] J. McNames Portland State University ECE 223 DT Fourier Series Ver

10 Picking Basis Functions We would like basis functions that Are mutually orthogonal < x k [n], x l [n] >= 0 for all k l Cause a simple and known response when applied to LTI systems: x k [n] y k [n] Are meaningful Complex sinusoids satisfy these last two properties, but are they mutually orthogonal? J. McNames Portland State University ECE 223 DT Fourier Series Ver

11 Example 1: Complex Sinusoidal Sum Consider the set of basis functions x k [n] = e jkωn where Ω 2π N 1. Are these basis functions orthogonal? 2. What are their norms? Hint: recall the finite sum formula N 1 n=0 a n = 1 an 1 a J. McNames Portland State University ECE 223 DT Fourier Series Ver

12 Example 1: Workspace J. McNames Portland State University ECE 223 DT Fourier Series Ver

13 Harmonically Related Complex Sinusoids x k [n] = e jkωn where Ω = 2π N Harmonically related complex sinusoids are very useful They form an orthogonal set of basis functions The response of an LTI system to each basis function is known e jkωn H(e jkω )e jkωn It is easy to solve for the coefficients for signals that can be represented as x[n] = X[k]e jkωn k J. McNames Portland State University ECE 223 DT Fourier Series Ver

14 DT Periodic Signals 1 x[n] x[n] = { 1.0 n = 3k 0.5 Otherwise n Suppose we have a DT signal x[n] that we know is periodic The signal is applied at the input of an LTI system We would like to estimate the signal as a sum of complex sinusoids ˆx[n] = k X[k] e jω kn J. McNames Portland State University ECE 223 DT Fourier Series Ver

15 DT Periodic Signals Design Task ˆx[n] = k X[k] e jω kn Here theˆsymbol is used to indicate that the sum is an approximation (estimate) of x[n] Enables us to calculate the system output easily Must pick The frequencies Ω k The range of the sum k The coefficients X[k] Even if we make good choices, how good will the estimate be? J. McNames Portland State University ECE 223 DT Fourier Series Ver

16 Picking the Frequencies ˆx[n] = k X[k] e jω kn We know x[n] is periodic with some fundamental period N If ˆx[n] is to approximate x[n] accurately, it should also repeat every N samples In order for ˆx[n] to be periodic with period N, every complex sinusoid must also be periodic Only a harmonic set of complex sinusoids have this property Thus Ω k = kω where Ω = 2π N ˆx[n] = k X[k] e jkωn J. McNames Portland State University ECE 223 DT Fourier Series Ver

17 Design Task: Picking the Range ˆx[n] = k X[k] e jkωn Recall that there are only N distinct complex sinusoids that repeat every N samples e jkωn = e j(k+ln)n where Ω = 2π N Thus we can pick the range of the sum so that it includes only these terms Typical choices ˆx[n] = N 1 k=0 X[k] e jkωn ˆx[n] = (N 1)/2 k= (N 1)/2 X[k] e jkωn J. McNames Portland State University ECE 223 DT Fourier Series Ver

18 Equivalent Expressions for Exponential Sums x[n] = k X[k] e jkωn Since only N of the harmonics are distinct, we may truncate the sum so that it only contains only N distinct terms. It does not matter which set of N terms are used. x[n] = N 1 X[k] e jkωn = N 1+l X[k] e jkωn = X[k] e jkωn k=0 k=l k=<n> Again, note the new notation: X[k] e jkωn N+l X[k] e jkωn for any l k=<n> k=1+l J. McNames Portland State University ECE 223 DT Fourier Series Ver

19 Design Task: Picking the Coefficients ˆx[n] = X[k] e jkωn MSE = 1 N x[n] ˆx[n] 2 k=<n> We would like to pick the coefficients X[k] so that ˆx[n] is as close to x[n] as possible But what is close? One measure of the difference between two signals is the mean squared error (MSE) There are other measures, but this is a convenient one because we can differentiate it Since the signal is periodic, the MSE is calculated over a single fundamental period of N consecutive samples How do we pick the coefficients X[k] to minimize the MSE? J. McNames Portland State University ECE 223 DT Fourier Series Ver

20 Design Task: Coefficient Optimization ˆx[n] = X[k] e jkωn MSE = 1 N x[n] ˆx[n] 2 k=<n> Solving for the optimal coefficients is difficult However, suppose that an optimal solution exists such that MSE = 0 If such a solution exists, we know the sum of errors will also be zero 0 = x[n] ˆx[n] J. McNames Portland State University ECE 223 DT Fourier Series Ver

21 Design Task: Solve for the Coefficients 0 = x[n]e jlωn ˆx[n]e jlωn x[n]e jlωn = x[n]e jlωn = = k=<n> = k=<n> = k=<n> = X[l]N ˆx[n]e jlωn ( k=<n> X[k] X[k] X[k] e jkωn ) X[k] (Nδ[k l]) e jkωn e jlωn e j(k l)ωn e jlωn J. McNames Portland State University ECE 223 DT Fourier Series Ver

22 Design Task: Optimal Coefficients Thus, the coefficient of the lth complex sinusoid that minimize the MSE is X[l] = 1 x[n]e jlωn N Note that this is the same result we would have obtained by taking an inner product with the basis function With more algebra, you should be able to show that these coefficients result in MSE = 0! Thus, ˆx[n] = x[n] This means any DT periodic signal can be exactly represented as a sum of N complex sinusoidal harmonics J. McNames Portland State University ECE 223 DT Fourier Series Ver

23 x[n] = DTFS Observations X[k] e jkωn X[k] = 1 N k=<n> x[n]e jkωn The first equation is called the synthesis equation The second equation is called the analysis equation The coefficients X[k] are called the spectral coefficients or Fourier series coefficients of x[n] We denote the relationship of x[n] and X[k] by x[n] FS X[k] Both are complete representations of the signal: If we know one, we can compute the other X[k] is a function of frequency (kω) J. McNames Portland State University ECE 223 DT Fourier Series Ver

24 Further DTFS Observations & Comments x[n] = X[k] e jkωn X[k] = 1 N x[n]e jkωn k=<n> The DTFS transform is special because both sums are finite This permits us to calculate the DTFS exactly with computers (e.g. MATLAB) and microprocessors Later we will see how the DTFS is used to compute the other 3 transforms There is also a very fast algorithm (FFT) for computing the DTFS The Fourier series representation of signals is useful for LTI system design and analysis because we know how LTI systems affect sinusoidal signals J. McNames Portland State University ECE 223 DT Fourier Series Ver

25 Example 2: Pulse x[n] 1 -N -m 0 m N n What type of symmetry does the signal have? Find the DT Fourier series coefficients. Plot the coefficient spectrum, partial sums of the Fourier series components, and the MSE versus number of coefficients. J. McNames Portland State University ECE 223 DT Fourier Series Ver

26 Example 2: Workspace J. McNames Portland State University ECE 223 DT Fourier Series Ver

27 Example 2: Coefficient Spectrum 0.5 Discrete Time Fourier Series Coefficients X[k] X[k] kth (harmonic) J. McNames Portland State University ECE 223 DT Fourier Series Ver

28 Example 2: Partial Fourier Series 1.2 Partial Fourier Series Approximation (k=0) Time (samples) J. McNames Portland State University ECE 223 DT Fourier Series Ver

29 Example 2: Partial Fourier Series 1.2 Partial Fourier Series Approximation (k= 1 to 1) Time (samples) J. McNames Portland State University ECE 223 DT Fourier Series Ver

30 Example 2: Partial Fourier Series 1.2 Partial Fourier Series Approximation (k= 2 to 2) Time (samples) J. McNames Portland State University ECE 223 DT Fourier Series Ver

31 Example 2: Partial Fourier Series 1.2 Partial Fourier Series Approximation (k= 3 to 3) Time (samples) J. McNames Portland State University ECE 223 DT Fourier Series Ver

32 Example 2: Partial Fourier Series 1.2 Partial Fourier Series Approximation (k= 4 to 4) Time (samples) J. McNames Portland State University ECE 223 DT Fourier Series Ver

33 Example 2: Partial Fourier Series 1.2 Fourier Series Synthesis (k= 5 to 5) Time (samples) J. McNames Portland State University ECE 223 DT Fourier Series Ver

34 Example 2: Mean Squared Error MSE Coefficients J. McNames Portland State University ECE 223 DT Fourier Series Ver

35 Example 2: MATLAB Code function [] = DTPulse(); %============================================================================== % User- Specified Parameters %============================================================================== N = 11; % Fundamental period ( samples) m = 2; % Pulse width = (1 + 2m) samples %============================================================================== % Preprocessing %============================================================================== w = 2*pi/N; n = (-15:15); t = (-15:0.01:15) ; tm = mod(n,n); %============================================================================== % Construct one period of the signal %============================================================================== x = 1*(tm m) + 1*(tm N-m); %============================================================================== % Plot the Coefficient Spectrum %============================================================================== figure(1); FigureSet(1, LTX ); K = -25:25; J. McNames Portland State University ECE 223 DT Fourier Series Ver

36 for c1 = 1:length(K), k = K(c1); if k==0, a(c1) = (1/N)*(m+1/2)/(1/2); else a(c1) = (1/N)*sin(w*k*(m+1/2))./sin(w*k/2); end; end; h = stem(k,a); set(h, Color,[ ]); set(h, Marker,. ); set(h, MarkerSize,10); set(h, LineWidth,1.0); xlabel( kth (harmonic) ); ylabel( X[k] ); title( Discrete-Time Fourier Series Coefficients X[k] ); AxisLines; box off; AxisSet(8); print -depsc DTPulseSpectrum; NP = 0:(N-1)/2; k = 1:N-1; a(1) = (1/N)*(m+1/2)/(1/2); a(k+1) = (1/N)*sin(w*k*(m+1/2))./sin(w*k/2); %============================================================================== % Plot the Approximations %============================================================================== MSE = zeros((n+1)/2,1); for c1 = 1:length(NP), J. McNames Portland State University ECE 223 DT Fourier Series Ver

37 figure; FigureSet(1, LTX ); hold on; xd = zeros(size(n)); xc = zeros(size(t)); for c2 = 1:c1, k = c2-1; if k==0, xpd = a(c2)*exp(j*k*w*n); xpc = a(c2)*exp(j*k*w*t); else xpd = a(c2)*exp(j*k*w*n) + conj(a(c2))*exp(-j*k*w*n); xpc = a(c2)*exp(j*k*w*t) + conj(a(c2))*exp(-j*k*w*t); end; h = plot(t,xpc); set(h, Color,[ ]); set(h, LineWidth,0.2); h = plot(n,xpd,. ); set(h, Color,[ ]); set(h, MarkerSize,6); xd = xd + xpd; xc = xc + xpc; end; MSE(c1) = mean((x(1:n)-xd(1:n)).^2); h = plot(t,xc, k ); set(h, LineWidth,0.4); h = stem(n,x, r ); J. McNames Portland State University ECE 223 DT Fourier Series Ver

38 set(h, Marker,. ); set(h, MarkerSize,11); set(h, LineWidth,1.4); h = stem(n,xd, b ); set(h, Marker,. ); set(h, MarkerSize,8); set(h, LineWidth,0.8); xlabel( Time (samples) ); if c1==1, st = sprintf( Partial Fourier Series Approximation (k=0) ); elseif c1==length(np), st = sprintf( Fourier Series Synthesis (k=%d to %d),-(c1-1),c1-1); else st = sprintf( Partial Fourier Series Approximation (k=%d to %d),-(c1-1),c1-1); end; title(st); xlim([min(t) max(t)]); ylim([ ]); AxisLines; box off; AxisSet(8); st = sprintf( print -depsc DTPulse%d,c1-1); eval(st); end; %============================================================================== % Plot the Mean Square Error %============================================================================== figure; J. McNames Portland State University ECE 223 DT Fourier Series Ver

39 FigureSet(1, LTX ); k = [1 1+2*(1+0:(N-1)/2)]; h = stem(k,mse, k ); set(h, Marker,. ); set(h, MarkerSize,11); set(h, LineWidth,1.4); xlabel( Coefficients ); ylabel( MSE ); box off; AxisSet(8); print - depsc DTPulseMSE; J. McNames Portland State University ECE 223 DT Fourier Series Ver

40 DTFS Terminology x[n] = X[k] e jkωn X[k] = 1 N x[n]e jkωn k=<n> X[k] is complex-valued in general The function X[k] is called the magnitude spectrum of x[n] The function arg{x[k]} is called the phase spectrum of x[n] J. McNames Portland State University ECE 223 DT Fourier Series Ver

41 Summary A finite sum of complex sinusoids can represent any DT periodic signal! Harmonic sets of complex sinusoids are special Orthogonal set Constant amplitude Eigenfunctions of LTI systems The spectral coefficients are useful for telling us where the power of the signal is concentrated in the frequency domain The coefficients of these sinusoids are useful for signal analysis, LTI system design and analysis J. McNames Portland State University ECE 223 DT Fourier Series Ver