Overview of Discrete-Time Fourier Transform Topics Handy equations and its Definition Low- and high- discrete-time frequencies Convergence issues DTFT of complex and real sinusoids Relationship to LTI systems DTFT of pulse signals DTFT of periodic signals Relationship to DT Fourier series Impulse trains in time and frequency a n = n= N n= N n=m Handy Equations a n = an a a, a < a n = am a N a na n = n= a a, a < a n = a(n+.5) a (N+.5) a.5 a.5 You should be able to prove all of these. J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 2 Handy Limits sin ( Ω(N ± 2 )) sin(ω 2 ) = + cos ( Ω(N ± 2 )) cos(ω 2 ) = ± cos ( Ω(N ± 2 )) sin(ω 2 ) = l= l= δ(ω l) δ(ω π l) sin ( Ω(N ± 2 )) cos(ω 2 ) = First is roughly analogous to a sinc function All are periodic functions of frequency Ω with fundamental period of Orthogonality Defined Two non-periodic power signals x [n] and x 2 [n] are orthogonal if and only if x [n]x 2N + 2[n] = J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 3 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 4
Orthogonality of Complex Sinusoids Consider two (possibly non-harmonic) complex sinusoids Are they orthogonal? 2N + x [n] =e jω n x 2 [n] =e jω 2n x [n]x 2[n] = 2N + = = 2N + { Ω Ω 2 =l Otherwise e jω n e jω 2n e j(ω Ω 2 )n Importance of Orthogonality Suppose that we know a signal is composed of a linear combination of non-harmonic complex sinusoids x[n] = π How do we solve for the coefficients? x[n]e jω on π = [ [ = π ] e jω on 2N + ] e jωn e jω on dω J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 5 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 6 = [ π = π [ = = = π π π = X(e jω o ) Workspace e j(ω Ω o)n ] dω e j(ω Ωo)(N+.5) e j(ω Ω ] o)(n+.5) dω e j(ω Ω o).5 e j(ω Ω o).5 [ ] X(e jω sin[(ω Ω o )(N +.5)] ) dω sin[(ω Ω o ).5] δ(ω Ω o ± l)dω l= δ(ω Ω o )dω Definition F{x[n]} = = F { } = x[n] = x[n]e jωn Denote relationship as x[n] FT Why use this odd notation for the transform? Wouldn t X(Ω) be simpler than? Answer: this awkward notation is consistent with the z-transform X(z) = x[n]z n =X(z) z=e jω This also enables us to distinguish between the DT & CT Fourier transforms J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 7 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 8
Mean Squared Error F{x[n]} = = ˆx[n] = MSE = x[n]e jωn x[n] ˆx[n] 2 Like the Fourier series, it can be shown that minimizes the MSE over all possible functions of Ω Like the DTFS, the error converges to zero Note: this isn t in the text = Observations x[n]e jωn x[n] = Called the analysis and synthesis equations, respectively Recall that e jωn = e j(ω+l)n, for any pair of integers l and n Thus, is a periodic function of Ω with a fundamental period of Unlike the DT Fourier series, the frequency Ω is continuous Thus the DT synthesis integral can be taken over any continuous interval of length J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 9 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 = Comments x[n]e jωn x[n] = describes the frequency content of the signal x[n] x[n] can be thought of as being composed of a continuum of frequencies represents the density of the component at frequency Ω. π.2 π.4 π.6 π.8 π. π Discrete-Time Harmonics Equivalence of Discrete Time Harmonics 8 6 4 2 2 4 6 8 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 2
MATLAB Code function [] = Harmonics(); close all; n = -:; t = -:.:; w = [:.2:]*pi; nw = length(w); FigureSet(, LTX ); for cnt = :length(w), subplot(nw,,cnt); h = plot([min(t) max(t)],[ ], k:,t,cos(t*w(cnt)), b,t,cos(t*(w(cnt)+2*pi)), r ); hold on; h = stem(n,cos(n*w(cnt))); set(h(), Marker,. ); set(h(), MarkerSize,5); set(h, Color, k ); hold off; ylabel(sprintf( %3.f \\pi,w(cnt)/pi)); y([-.5.5]); if cnt==, title( Equivalence of Discrete-Time Harmonics ); end; if cnt~=nw, set(gca, XTickLabel, ); end; end; AxisSet(8); print -depsc Harmonics; Discrete-Time Frequency Concepts Recall that e j(ω+l)n =e jωn If seemingly very high-frequency discrete-time signals, cos ((Ω + l)n), are equal to low-frequency discrete-time signals, cos(ωn), what does low- and high-frequency mean in discrete-time? Note that the units of Ω are radians per sample A sinusoid with a frequency of. radians per sample is the same as one with a frequency of (.+) radians per sample Recall that cos(πn) =( ) n No DT signal can oscillate faster between two samples No DT signal can oscillate slower than radians per sample Thus Ω = π = l(π + ) is the highest perceivable DT frequency Ω==l() is the lowest perceivable frequency J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 3 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 4 Discrete-Time Frequency Concepts Continued Example 4: Unit Impulse Find the Fourier transform of x[n] =δ[n]. 4π 3π π 3π 4π Ω 4π 3π π 3π 4π Ω Low frequencies are those that are near High frequencies are those near ±π Intermediate frequencies are those in between Note that the highest frequency, π radians per sample is equal to.5 cycles per sample We will encounter this concept again when we discuss sampling J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 5 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 6
= Convergence x[n]e jωn x[n] = Sufficient conditions for the convergence of the discrete-time Fourier transform of a bounded discrete-time signal: (any one of the following are sufficient) Finite duration: There exists an N such that x[n] =for n >N Absolutely summable: x[n] < x[n] 2 < Finite energy: The synthesis equation always converges There is no Gibb s phenomenon in the time domain Example 5: Inverse of Impulse Train Sketch the following impulse train and find the inverse Fourier transform. = δ(ω Ω l) l= J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 7 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 8 Example 5: Workspace Example 6: Constant Find the Fourier transform of x[n] =. J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 9 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 2
Example 6: Workspace = Convergence Revisited x[n]e jωn x[n] = The DTFT is said to converge if is finite for all Ω We ve now seen two examples where the DTFT was infinite at specific frequencies In these cases the DTFT didn t converge But we were able to represent the transform with impulses δ(ω Ω ) anyway The use of impulses enables us to represent signals that contain periodic elements including constants and sinusoids J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 2 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 22 Fourier Transform & Transfer Functions M Y (e jω k= )= b ke jωk + N k= a ke jωk X(ejΩ )=H(e jω ) The time-domain relationship of y[n] and x[n] can be complicated In the frequency domain, the relationship of Y (e jω ) to of LTI systems described by difference equations simplifies to a rational function of e jω The numerator/denomenator sums are not Fourier series For real systems, H(e jω ) is usually a rational ratio of two polynomials H(e jω ) is the discrete-time transfer function Specifically, the transfer function of an LTI system can be defined as the ratio of Y (e jω ) to Same story as the continuous-time case Example 7: Relationship to DT LTI Systems Suppose the impulse response h[n] is known for an LTI DT system. Derive the relationship between a sinusoidal input signal and the output of the system. J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 23 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 24
Example 7: Workspace Example 8: Decaying Exponential Find the Fourier transform of h[n] =a n u[n] where a <. Sketchthe transform over a range of 3π to 3π for a =.5 and a =.5. J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 25 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 26 Example 8: Workspace Example 9: First-Order Filter a =.5 x[n] X(e jω ) ( o ).5 Fourier Transform of (.5) n u[n].5 8 6 4 2 2 4 6 8 2 X(e jω ) 9.4248 5 6.2832 3.46 3.46 6.2832 9.4248 5 9.4248 6.2832 3.46 3.46 Frequency (rad/sample) 6.2832 9.4248 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 27 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 28
x[n] X(e jω ) ( o ).5 Example : First-Order Filter a =.5 Fourier Transform of (.5) n u[n].5 8 6 4 2 2 4 6 8 2 X(e jω ) 9.4248 5 6.2832 3.46 3.46 6.2832 9.4248 5 9.4248 6.2832 3.46 3.46 Frequency (rad/sample) 6.2832 9.4248 Example : MATLAB Code %function [] = FirstOrder(); close all; w = -3*pi:6*pi/:3*pi; n = -8:8; figure; FigureSet(, LTX ); a =.5; x = (a.^n).*(n>=); X =./(-a*exp(-j*w)); Y =./(exp(-j*w/2).*2.*sin(w/2)); subplot(3,,); h = stem(n,x, b ); set(h(), MarkerFaceColor, b ); set(h(), LineWidth,.); set(h(), MarkerSize,2); set(h(2), LineWidth,.6); set(h(3), Visible, Off ); ylabel( x[n] ); title(sprintf( Fourier Transform of (%3.f)^n u[n],a)); x([min(n) max(n)]); y([-.55.6]); subplot(3,,2); h = plot(w,abs(x), r ); set(h, LineWidth,.6); ylabel( X(e^{j\omega}) ); x([-3*pi 3*pi]); y([ 2.2]); set(gca, XTick,[-3:3]*pi); grid on; J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 29 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 3 subplot(3,,3); h = plot(w,angle(x)*8/pi, r ); set(h, LineWidth,.6); ylabel( \angle X(e^{j\omega}) (^o) ); xlabel( Frequency (rad/sample) ); x([-3*pi 3*pi]); set(gca, XTick,[-3:3]*pi); grid on; AxisSet(8); print -depsc FirstOrderLowpass; figure; FigureSet(, LTX ); a = -.5; x = (a.^n).*(n>=); X =./(-a*exp(-j*w)); subplot(3,,); h = stem(n,x, b ); set(h(), MarkerFaceColor, b ); set(h(), LineWidth,.); set(h(), MarkerSize,2); set(h(2), LineWidth,.6); set(h(3), Visible, Off ); ylabel( x[n] ); title(sprintf( Fourier Transform of (%3.f)^n u[n],a)); x([min(n)-.5 max(n)+.5]); y([-.55.6]); subplot(3,,2); h = plot(w,abs(x), r ); set(h, LineWidth,.6); ylabel( X(e^{j\omega}) ); x([-3*pi 3*pi]); y([ 2.2]); set(gca, XTick,[-3:3]*pi); grid on; subplot(3,,3); h = plot(w,angle(x)*8/pi, r ); set(h, LineWidth,.6); ylabel( \angle X(e^{j\omega}) (^o) ); xlabel( Frequency (rad/sample) ); x([-3*pi 3*pi]); set(gca, XTick,[-3:3]*pi); grid on; AxisSet(8); print -depsc FirstOrderHighpass; J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 3 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 32
Example : Pulse Find the Fourier transform of the following pulse signal. Sketch the transform over a range of 3π to 3π for N =5,, & and N =8. { n N p N [n] = n >N Example : Workspace J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 33 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 34 Example 2: Pulse Transform for N =5 Fourier Transform of p 5 [n] 8 Example 3: Pulse Transform for N = Fourier Transform of p [n] 2 5 P(e jω ) 6 4 2 P(e jω ) 5 2 9.4248 6.2832 3.46 3.46 6.2832 9.4248 9.4248 6.2832 3.46 3.46 6.2832 9.4248 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 35 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 36
Example 4: Pulse Transform for N = Example 4: MATLAB Code P(e jω ) 2 5 5 Fourier Transform of p [n] function [] = Pulse(); close all; s = e-4; w = s:s:3*pi; w = [-w(length(w):-:) w]; N = [5 ]; for c=:length(n), figure; FigureSet(, LTX ); X = sin(w*(n(c)+.5))./sin(w/2); h = plot(w,x, LineWidth,.4); ylabel( P(e^{j\omega}) ); title(sprintf( Fourier Transform of p_{%d}[n],n(c))); x([-3*pi 3*pi]); y(.5*[min(x) max(x)]); set(gca, XTick,[-3:3]*pi); grid on; AxisSet(8); print( -depsc,sprintf( Pulse%3d,N(c))); end; 9.4248 6.2832 3.46 3.46 6.2832 9.4248 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 37 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 38 Example 5: Periodic Discrete Time Signals Find the Fourier transform of the following periodic signal. x[n] = X[k]e jkω on k=<n> Example 5: Workspace where Ω= N for some integer N. J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 39 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 4
Example 6: Relationship to Fourier Series Suppose that we have a periodic signal x[n] with fundamental period N. Define the truncated signal x[n] as follows. { x[n] n + n n + N x[n] = otherwise Example 6: Workspace Determine how the Fourier transform of x[n] is related to the discrete-time Fourier series coefficients of x[n] Recall that X[k] = N n=<n> x[n]e jk(/n)n J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 4 J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 42 Summary of Key Concepts is a periodic function of Ω with a fundamental period of Two discrete-time complex exponentials with frequencies that differ by a multiple of are equal: e jωn =e j(ω+l)n The highest perceivable discrete-time frequency is π radians per sample The lowest perceivable discrete-time frequency is radians per sample The analysis equation may or may not converge, the synthesis equation always converges Quasi-periodic signals have a DTFT that consists of impulses at multiples of the fundamental frequency The DTFT enables us to analyze and understand systems described by difference-equations more thoroughly J. McNames Portland State University ECE 223 DT Fourier Transform Ver..23 43