Overview of Complex Sinusoids Topics Eigenfunction & eigenvalues - PowerPoint PPT Presentation

Overview of Complex Sinusoids Topics • Eigenfunction & eigenvalues of LTI systems • Understanding complex sinusoids • Four classes of signals • Periodic signals • CT & DT Exponential harmonics J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 1

Complex Sinusoids • A complex sinusoid is defined as x ( t ) = Ae jωt = A [cos( ωt ) + j sin( ωt )] • These are a special case of complex exponentials x ( t ) = Ae st = Ae αt [cos( ωt ) + j sin( ωt )] when α = 0 J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 2

Review: Signals as Impulses h ( t ) h [ n ] x ( t ) y ( t ) x [ n ] y [ n ] • Fourier series is used for signal decomposition • In ECE 222 we decomposed signals into sums and of impulses � ∞ ∞ � x ( t ) = x ( τ ) δ ( t − τ ) d τ x [ n ] = x [ k ] δ [ n − k ] −∞ k = −∞ � ∞ ∞ � y ( t ) = x ( τ ) h ( t − τ ) d τ y [ n ] = x [ k ] h [ n − k ] −∞ k = −∞ • We then used the LTI properties (linearity and time invariance) to solve for the output as a sum of the impulse responses J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 3

Review of Energy and Power Signals � ∞ � T 1 | x ( t ) | 2 d t | x ( t ) | 2 d t E ∞ = P ∞ = lim 2 T T →∞ −∞ − T ∞ N 1 � � | x [ n ] | 2 | x [ n ] | 2 E ∞ = P ∞ = lim 2 N + 1 N →∞ n = −∞ n = − N • A signal is an energy signal if E ∞ < ∞ • A signal is a power signal if 0 < P ∞ < ∞ • Most signals are either energy signals or power signals • A signal cannot be both J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 4

Orthogonalality A pair of real-valued energy signals are orthogonal if � ∞ x 1 ( t ) x ∗ 0 = 2 ( t ) d t −∞ ∞ � x 1 [ n ] x ∗ 0 = 2 [ n ] n = −∞ A pair of real-valued power signals are orthogonal if � T 1 x 1 ( t ) x ∗ 0 = lim 2 ( t ) d t 2 T T →∞ − T N 1 � x 1 [ n ] x ∗ 0 = lim 2 [ n ] 2 N + 1 N →∞ n = − N J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 5

Examples of Orthogonal Signals Like impulses, complex sinusoids are special • Impulses are orthogonal to one another � ∞ δ ( t − t 1 ) δ ( t − t 2 ) d t = 0 for t 1 � = t 2 −∞ ∞ � δ [ n − n 1 ] δ [ n − n 2 ] = 0 for n 1 � = n 2 n = −∞ • Complex sinusoids are also orthogonal to one another � T 1 e jω 1 t e jω 2 t d t = 0 for ω 1 � = ω 2 lim 2 T T →∞ − T N 1 e j Ω 1 n e j Ω 2 n = 0 for Ω 1 � = Ω 2 � lim 2 N + 1 N →∞ n = − N J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 6

Importance of Orthogonality • Orthogonal basis functions (e.g., complex sinusoids, shifted impulses) enable us to decompose signals into distinct (orthogonal) components • More later J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 7

Eigenfunctions & Eigenvalues h ( t ) h [ n ] x ( t ) y ( t ) x [ n ] y [ n ] • There are other basic signals that are also orthogonal • But exponentials have another special property: • You may be familiar with eigenvectors & eigenvalues for matrices • There is a related concept for LTI systems • Any signal x ( t ) or x [ n ] that is only scaled when passed through a system is called an eigenfunction of the system – y ( t ) = x ( t ) ∗ h ( t ) = c x ( t ) – y [ n ] = x [ n ] ∗ h [ n ] = c x [ n ] • The scaling constant c is called the system’s eigenvalue • Complex exponentials are eigenfunctions of LTI systems • Complex sinusoids are the only eigenfunctions of LTI systems that have finite power! J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 8

CT LTI System Response to Complex Exponentials Let x ( t ) = e jωt . Then � ∞ y ( t ) = h ( τ ) x ( t − τ ) d τ −∞ � ∞ h ( τ ) e jω ( t − τ ) d τ = −∞ � ∞ h ( τ ) e jωt e − jωτ d τ = −∞ � ∞ h ( τ )e − jωτ d τ = e jωt −∞ = e jωt H ( jω ) where � ∞ h ( τ )e − jωτ d τ H ( jω ) = −∞ J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 9

DT LTI System Response to Complex Exponentials Let x [ n ] = e j Ω n ∞ � y [ n ] = h [ k ] x [ n − k ] k = −∞ ∞ � h [ k ] e j Ω( n − k ) = k = −∞ ∞ � h [ k ] e j Ω n e − j Ω k = k = −∞ ∞ � = e j Ω n h [ k ] e − j Ω k k = −∞ = e j Ω n H (e j Ω ) where ∞ � H (e j Ω ) = h [ k ] e − j Ω k k = −∞ J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 10

Eigenfunctions & Eigenvalues h ( t ) h [ n ] x ( t ) y ( t ) x [ n ] y [ n ] • Thus, complex sinusoids are eigenfunctions of LTI systems e jωt → H ( jω )e jωt e j Ω n → H (e j Ω n )e j Ω n • The Fourier transform of the impulse response are the eigenvalues � ∞ h ( t )e − jωt d t H ( jω ) = F { h ( t ) } = −∞ ∞ � H (e j Ω ) = F { h [ n ] } = h [ n ] e − j Ω n n = −∞ • H ( jω ) and H (e j Ω ) are functions of frequency • Called the frequency response of the system J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 11

Example 1: Real and Complex Sinusoids Are real sinusoids eigenfunctions of LTI systems? J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 12

Magnitude and Phase Response H (e j Ω ) x ( t ) y ( t ) x [ n ] y [ n ] H ( jω ) � ∞ h ( t )e − jωt d t H ( jω ) = F { h ( t ) } = −∞ ∞ � H (e j Ω ) = F { h [ n ] } = h [ n ] e − j Ω n n = −∞ • Both the eigenfunctions and eigenvalues of LTI systems are complex-valued • | H ( jω ) | and | H (e j Ω ) | are called the magnitude response • The complex phase angle of H ( jω ) and H (e j Ω ) – Called the phase response – Denoted as arg H ( jω ) in the text � � Im { H ( jω ) } – This is not the same as arctan , in general Re { H ( jω ) } J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 13

Magnitude and Phase Response Continued H (e j Ω ) x ( t ) H ( jω ) y ( t ) x [ n ] y [ n ] e jωt → | H ( jω ) | e j arg H ( jω ) e jωt = | H ( jω ) | e j ( ωt +arg H ( jω )) e j Ω n → | H (e j Ω ) | e j arg H (e j Ω ) e j Ω n = | H (e j Ω ) | e j ( ωt +arg H (e j Ω )) Thus, if a complex sinusoid is applied at the input to an LTI system • The system scales the amplitude by | H ( · ) | • The system changes the phase by arg H ( · ) J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 14

Sums of Complex Exponentials h ( t ) h [ n ] x ( t ) y ( t ) x [ n ] y [ n ] • Fourier series represent signals as sums (or integrals) of complex sinusoids N 0 − 1 ∞ � � a k e jk Ω 0 n a k e jkω 0 t x ( t ) = x [ n ] = k = −∞ k =0 • Since the signals are real, the imaginary portions of the complex exponentials must cancel out to zero • This decomposition makes it particularly easy to solve for and analyze the system output • Sums of complex sinusoids can only represent periodic and nearly periodic signals • Integrals of complex sinusoids are more general J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 15

Complex Exponential Sums H (e j Ω ) x ( t ) H ( jω ) y ( t ) x [ n ] y [ n ] By linearity and time-invariance (LTI), � � a k e jω k t a k H ( jω k )e jω k t x ( t ) = → y ( t ) = k k � � a k e j Ω k n a k H (e j Ω k )e j Ω k n x [ n ] = → y [ n ] = k k • Thus if the input signal can be expressed as a sum of complex sinusoids, so can the output of the LTI system • But what types of signals can be represented in this form? • Virtually all of the (periodic) signals that we are interested in! • Important and interesting idea J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 16

Continuous-Time Signal Intuition � � a k e jω k t a k H ( jω k )e jω k t x ( t ) = → y ( t ) = k k • Fourier transforms represent signals as sums (or integrals) of complex sinusoids • It is therefore worthwhile to understand complex sinusoids as thoroughly as possible s = jω = e jωt = cos( ωt ) + j sin( ωt ) x ( t ) = e st � � J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 17

Example 2: x ( t ) = e jt 1 0.5 Real Part 0 −0.5 −1 −10 −8 −6 −4 −2 0 2 4 6 8 10 1 Imaginary Part 0.5 0 −0.5 −1 −10 −8 −6 −4 −2 0 2 4 6 8 10 Time (s) J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 18

Example 2: x ( t ) = e jt Complex:Blue Real:Green Imaginary:Red Complex Plane:Yellow 1 0.5 Imaginary Part 0 −0.5 −1 −10 −1 −5 −0.5 0 0 5 0.5 Time (s) 1 10 Real Part J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 19

Example 2: MATLAB Code w = j*1; fs = 500; % Sample rate (Hz) t = -10:1/fs:10; % Time index (s) y = exp(w*t); N = length(t); subplot(2,1,1); h = plot(t,real(y)); box off; grid on; ylim([-1.1 1.1]); ylabel(’Real Part’); subplot(2,1,2); h = plot(t,imag(y)); box off; grid on; ylim([-1.1 1.1]); xlabel(’Time (s)’); ylabel(’Imaginary Part’); J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 20