Overview of Discrete-Time Fourier Transform Topics Handy equations - PowerPoint PPT Presentation

Overview of Discrete-Time Fourier Transform Topics • Handy equations and limits • 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 J. McNames Portland State University ECE 223 DT Fourier Transform Ver. 1.23 1

Handy Equations ∞ 1 � a n = | a | < 1 1 − a, n =0 N − 1 1 − a N � a n = 1 − a n =0 N − 1 a M − a N � a n = 1 − a n = M ∞ a � na n = | a | < 1 1 − a, n =0 N a ( N +0 . 5) − a − ( N +0 . 5) � a n = a 0 . 5 − a − 0 . 5 n = − N You should be able to prove all of these. J. McNames Portland State University ECE 223 DT Fourier Transform Ver. 1.23 2

Handy Limits + ∞ Ω( N ± 1 � � sin 2 ) � lim = +2 π δ (Ω − 2 πℓ ) sin(Ω 1 2 ) N →∞ ℓ = −∞ + ∞ Ω( N ± 1 � � cos 2 ) � lim = ± 2 π δ (Ω − π − 2 πℓ ) cos(Ω 1 2 ) N →∞ ℓ = −∞ Ω( N ± 1 � � cos 2 ) lim = 0 sin(Ω 1 2 ) N →∞ Ω( N ± 1 � � sin 2 ) lim = 0 cos(Ω 1 2 ) N →∞ • First is roughly analogous to a sinc function • All are periodic functions of frequency Ω with fundamental period of 2 π J. McNames Portland State University ECE 223 DT Fourier Transform Ver. 1.23 3

Orthogonality Defined Two non-periodic power signals x 1 [ n ] and x 2 [ n ] are orthogonal if and only if N 1 � x 1 [ n ] x ∗ lim 2 [ n ] = 0 2 N + 1 N →∞ n = − N J. McNames Portland State University ECE 223 DT Fourier Transform Ver. 1.23 4

Orthogonality of Complex Sinusoids Consider two (possibly non-harmonic) complex sinusoids x 1 [ n ] = e j Ω 1 n x 2 [ n ] = e j Ω 2 n Are they orthogonal? N N 1 1 � � e j Ω 1 n e − j Ω 2 n x 1 [ n ] x ∗ lim 2 [ n ] = lim 2 N + 1 2 N + 1 N →∞ N →∞ n = − N n = − N N 1 � e j (Ω 1 − Ω 2 ) n = lim 2 N + 1 N →∞ n = − N � 1 Ω 1 − Ω 2 = 2 πℓ = 0 Otherwise J. McNames Portland State University ECE 223 DT Fourier Transform Ver. 1.23 5

Importance of Orthogonality Suppose that we know a signal is composed of a linear combination of non-harmonic complex sinusoids � π x [ n ] = 1 X (e j Ω ) e j Ω n dΩ 2 π − π How do we solve for the coefficients X (e j Ω ) ? N � x [ n ]e − j Ω o n lim N →∞ n = − N � 1 � π N � X (e j Ω ) e j Ω n dΩ � e − j Ω o n = lim 2 π N →∞ − π n = − N � π � N � = 1 1 � X (e j Ω ) e j Ω n e − j Ω o n lim dΩ 2 π 2 N + 1 N →∞ − π n = − N J. McNames Portland State University ECE 223 DT Fourier Transform Ver. 1.23 6

Workspace � π N � � = 1 � X (e j Ω ) e j (Ω − Ω o ) n lim dΩ 2 π N →∞ − π n = − N � π e j (Ω − Ω o )( N +0 . 5) − e − j (Ω − Ω o )( N +0 . 5) � � = 1 X (e j Ω ) lim dΩ e j (Ω − Ω o )0 . 5 − e − j (Ω − Ω o )0 . 5 2 π N →∞ − π � π = 1 � sin[(Ω − Ω o )( N + 0 . 5)] � X (e j Ω ) lim dΩ 2 π sin[(Ω − Ω o )0 . 5] N →∞ − π � π ∞ = 1 � X (e j Ω ) 2 π δ (Ω − Ω o ± 2 πℓ ) dΩ 2 π − π ℓ = −∞ � π X (e j Ω ) δ (Ω − Ω o ) dΩ = − π = X (e j Ω o ) J. McNames Portland State University ECE 223 DT Fourier Transform Ver. 1.23 7

Definition + ∞ � X (e j Ω ) = x [ n ] e − j Ω n F { x [ n ] } = n = −∞ x [ n ] = 1 � X (e j Ω ) e j Ω n dΩ F − 1 � X (e j Ω ) � = 2 π 2 π FT ⇒ X (e j Ω ) • Denote relationship as x [ n ] ⇐ • Why use this odd notation for the transform? • Wouldn’t X (Ω) be simpler than X (e j Ω ) ? • Answer: this awkward notation is consistent with the z -transform + ∞ � x [ n ] z − n X (e j Ω ) = X ( z ) | z =e j Ω X ( z ) = n = −∞ • This also enables us to distinguish between the DT & CT Fourier transforms J. McNames Portland State University ECE 223 DT Fourier Transform Ver. 1.23 8

Mean Squared Error + ∞ � X (e j Ω ) = x [ n ] e − j Ω n F { x [ n ] } = n = −∞ 1 � X (e j Ω ) e j Ω n dΩ x [ n ] ˆ = 2 π 2 π + ∞ x [ n ] | 2 � MSE = | x [ n ] − ˆ n = −∞ • Like the Fourier series, it can be shown that X (e j Ω ) minimizes the MSE over all possible functions of Ω • Like the DTFS, the error converges to zero • Note: this isn’t in the text J. McNames Portland State University ECE 223 DT Fourier Transform Ver. 1.23 9

Observations + ∞ x [ n ] = 1 � X (e j Ω ) e j Ω n dΩ � X (e j Ω ) = x [ n ]e − j Ω n 2 π 2 π n = −∞ • Called the analysis and synthesis equations, respectively • Recall that e j Ω n = e j (Ω+ ℓ 2 π ) n , for any pair of integers ℓ and n • Thus, X (e j Ω ) is a periodic function of Ω with a fundamental period of 2 π • Unlike the DT Fourier series, the frequency Ω is continuous • Thus the DT synthesis integral can be taken over any continuous interval of length 2 π J. McNames Portland State University ECE 223 DT Fourier Transform Ver. 1.23 10

Comments + ∞ x [ n ] = 1 � X (e j Ω ) e j Ω n dΩ � X (e j Ω ) = x [ n ]e − j Ω n 2 π 2 π n = −∞ • X (e j Ω ) describes the frequency content of the signal x [ n ] • x [ n ] can be thought of as being composed of a continuum of frequencies • X (e j Ω ) represents the density of the component at frequency Ω J. McNames Portland State University ECE 223 DT Fourier Transform Ver. 1.23 11

Discrete-Time Harmonics Equivalence of Discrete−Time Harmonics 1 0.0 π 0 −1 1 0.2 π 0 −1 1 0.4 π 0 −1 1 0.6 π 0 −1 1 0.8 π 0 −1 1 1.0 π 0 −1 −10 −8 −6 −4 −2 0 2 4 6 8 10 J. McNames Portland State University ECE 223 DT Fourier Transform Ver. 1.23 12

MATLAB Code function [] = Harmonics(); close all; n = -10:10; t = -10:0.01:10; w = [0:0.2:1]*pi; nw = length(w); FigureSet(1,’LTX’); for cnt = 1:length(w), subplot(nw,1,cnt); h = plot([min(t) max(t)],[0 0],’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(1),’Marker’,’.’); set(h(1),’MarkerSize’,5); set(h,’Color’,’k’); hold off; ylabel(sprintf(’%3.1f \\pi’,w(cnt)/pi)); ylim([-1.05 1.05]); box off; if cnt==1, title(’Equivalence of Discrete-Time Harmonics’); end; if cnt~=nw, set(gca,’XTickLabel’,’’); end; end; AxisSet(8); print -depsc Harmonics; J. McNames Portland State University ECE 223 DT Fourier Transform Ver. 1.23 13

Discrete-Time Frequency Concepts • Recall that e j (Ω+ ℓ 2 π ) n = e j Ω n • If seemingly very high-frequency discrete-time signals, cos ((Ω + ℓ 2 π ) n ) , are equal to low-frequency discrete-time signals, cos(Ω n ) , 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 0.1 radians per sample is the same as one with a frequency of ( 0 . 1 + 2 π ) radians per sample • Recall that cos( πn ) = ( − 1) n • No DT signal can oscillate “faster” between two samples • No DT signal can oscillate “slower” than 0 radians per sample • Thus – Ω = π = ℓ ( π + 2 π ) is the highest perceivable DT frequency – Ω = 0 = ℓ (2 π ) is the lowest perceivable frequency J. McNames Portland State University ECE 223 DT Fourier Transform Ver. 1.23 14

Discrete-Time Frequency Concepts Continued X (e j Ω ) 1 Ω − 4 π − 3 π − 2 π − π π 0 2 π 3 π 4 π X (e j Ω ) 1 Ω − 4 π − 3 π − 2 π − π π 0 2 π 3 π 4 π • Low frequencies are those that are near 0 • High frequencies are those near ± π • Intermediate frequencies are those in between • Note that the highest frequency, π radians per sample is equal to 0.5 cycles per sample • We will encounter this concept again when we discuss sampling J. McNames Portland State University ECE 223 DT Fourier Transform Ver. 1.23 15

Example 4: Unit Impulse Find the Fourier transform of x [ n ] = δ [ n ] . J. McNames Portland State University ECE 223 DT Fourier Transform Ver. 1.23 16

Convergence + ∞ x [ n ] = 1 � X (e j Ω )e j Ω n dΩ � X (e j Ω ) = x [ n ]e − j Ω n 2 π 2 π 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 ] = 0 for | n | > N ∞ � – Absolutely summable: | x [ n ] | < ∞ ∞ n = −∞ | x [ n ] | 2 < ∞ � – Finite energy: n = −∞ • The synthesis equation always converges • There is no Gibb’s phenomenon in the time domain J. McNames Portland State University ECE 223 DT Fourier Transform Ver. 1.23 17

Example 5: Inverse of Impulse Train Sketch the following impulse train and find the inverse Fourier transform. ∞ � X (e j Ω ) = 2 π δ (Ω − Ω 0 − 2 πℓ ) ℓ = −∞ J. McNames Portland State University ECE 223 DT Fourier Transform Ver. 1.23 18

Example 5: Workspace J. McNames Portland State University ECE 223 DT Fourier Transform Ver. 1.23 19

Example 6: Constant Find the Fourier transform of x [ n ] = 1 . J. McNames Portland State University ECE 223 DT Fourier Transform Ver. 1.23 20