Causality, Linearity, and Time-Invariance in Signal Processing

ECE 8443 Pattern Recognition EE 3512 Signals: Continuous and Discrete LECTURE 17: FOURIER ANALYSIS REVISITED Objectives: Causality Linearity Time Invariance Temporal Models Response to Periodic Signals Ideal Filters Nonlinearities Resources: Wiki: The RC Circuit CN: Response of an RC Circuit CNX: Ideal Filters URL:

Causality A system is causal if the output does not depend on future values of the input, i.e., if the output at any time depends only on values of the input up to that time. All real-time physical systems are causal, because time only moves forward. Effect occurs after cause. (Imagine if you own a noncausal system whose output depends on tomorrow s stock price.) Causality does not apply to spatially varying signals. (We can move both left and right, up and down.) Causality does not apply to systems processing recorded signals, e.g. taped sports games vs. live broadcast. Examples: ( t y More examples: 1 ) ( = t y = = ) ( ) [ ] [ ] y t x t x n x n + + ( ) cos( ) x t t = ) 1 = ] 1 ( ) ( [ ] [ x t y n x n = ( ) ( ) y t x t e d = ) 1 + = ] 1 + ( ) ( [ ] [ y t x t y n x n 0 = = 2 2 ( ) ( ) [ ] [ ] y t x t y n x n = + ( ) sin( ) ( ) y t t x d 1 1 ( ) 1 = = ] 1 3 t 3 n ( ) ( ) [ ] [ ] [ y t x t y n x n 2 2 1 = ] 1 + + [ ] ( [ [ 1 ]) y n x n x n 2 EE 3512: Lecture 17, Slide 1

Linearity A system is linear if it obeys the principle of superposition: x1(t) y1(t) and x2(t) y2(t) xk[n] yk[n] k n x ] [ If: k k a x1(t) + b x2(t) a y1(t) + b y2(t) Then: [ ] a a y n k k k Question: Which of these systems are linear? y y y x x x = + t ( ) ( ) cos( ) y t x t EE 3512: Lecture 17, Slide 2

Time-Invariance Informally, a system is time-invariant (TI) if its behavior does not depend on the choice of t = 0. Then two identical experiments will yield the same results, regardless the starting time. Mathematically (in DT): A system is time-invariant (TI) if for any input x[n] and any time shiftn0, if x[n] y[n], then x[n n0] y[n n0]. Similarly for a CT time-invariant system,if x(t) y(t), then x(t t0) y(t t0). Examples: More examples: = = ( ) ( ) [ ] [ ] y t x t x n x n = + + ( ) 1 ( ) cos( ) y t x t t = ) 1 = ] 1 ( ) ( [ ] [ y t x t y n x n = ( ) ( ) y t x t e d = ) 1 + = ] 1 + ( ) ( [ ] [ y t x t y n x n 0 = = 2 2 ( ) ( ) y t x t [ ] [ ] y n x n = + ( ) sin( ) ( ) y t t x d 1 1 ( ) 1 = = ] 1 3 t 3 n ( ) ( ) y t x t [ ] [ ] [ y n x n 2 2 1 = ] 1 + + [ ] ( [ [ 1 ]) y n x n x n 2 EE 3512: Lecture 17, Slide 3

Response of a CT LTI System CT LTI (t h = ( ) ( ) * ( ) y t x t h t (t ) x ) Denote the system impulse response, h(t), as the output produced when the input is a unit impulse function, (t). From time-invariance: ( ) ( ) t h t From linearity: = = = ( ) ( ) ( ) ( ) ( ) ( ) ( ) * ( ) x t x t d y t x h t d x t h t This is referred to as the convolution integral for CT signals and systems. Its computation is completely analogous to the DT version: EE 3512: Lecture 17, Slide 4

Response of an LTI System to a Sinusoid Consider an LTI CT system with impulse response h(t): = = ( d ( ) ( ) * ( ) ) ( ) y t h t x t h x t We will assume that the Fourier transform of h(t) exists: j ( = t ) ( ) H h t e dt The output can be computed using our Fourier transform properties: ( ) ( and ) ( ) ( ) ( H Y X H Y = = = + ) ( ) ( ) ( ) ( ) X Y H X Suppose the input is a sinusoid: cos( ) ( 0 + = t A t x ) Using properties of the Fourier transform, we can compute the output: ( ) ( ) ( ) ( ) ( e e H A + + = ) j j = + + ( ) X A e e 0 0 = Y H X ( ) ( ) j j = + + ( ) 0 0 ( ) ( ) j j ( ) ( + ) A H e H e + 0 0 0 0 ( ) ( ) ( ) ( ) j j = ( ) ( ) j H j H ( ) A H e e e e 0 0 0 0 + 0 )( ) )( ) ( ( + + = + ( ) ( ) j H j H ( ) A H e e 0 0 0 0 0 ( ) ( ) ) 0 = = + + 1 - F ( ) ( ) ( ) cos ( y t Y A H t H 0 0 EE 3512: Lecture 17, Slide 5

Response To Periodic Inputs We can extend our example to all periodic signals using the Fourier series: ( ) variant (a cos ) ( 1 = k 0 = + + of trigonome the Fourier tric series) x t a A k t 0 k k The output of an LTI system is: ( ) + = 0 0 ) ( H a t y ( ) ( ( ) ) k + + cos A H k k t H k 0 0 0 k k 1 We can write the Fourier series for the output as: ( cos ) ( 1 k = ) = + k + y y y y t a A t 0 0 k k where, = = k = + k y 0 x y x k y x a ) 0 ( H ( ) ( ) a A A H H 0 0 0 k k k also, 1 = k = + k y k y k x k x c ( ) and c ( ) A H H 0 0 k 2 It is important to observe that since the spectrum of a periodic signal is a line spectrum, the output spectrum is simply a weighted version of the input, where the weights are found by sampling of the frequency response of the LTI system at multiples of the fundamental frequency, 0. EE 3512: Lecture 17, Slide 6

Response to Nonperiodic Inputs We can recover the output in the time domain using the inverse transform: 2 1 j j j = d t ( ) ( ) ( ) y t H e X e e These integrals are often hard to compute, so we try to circumvent them using transform tables and combinations of transform properties. Consider the response of our RC circuit to a single pulse: ) 2 / sin( ) ( = j X e ( ) 2 / / 1 RC / 1 + = j ( ) H e j RC = j j j ( ) ( ) ( ) Y e X sin( e H ) 2 / e / 1 RC / 1 + = ( ) 2 / j RC MATLAB code for the frequency response: RC=1; w=-40:.3:40; X=2*sin(w/2)./w; H=(1/RC)./(j*w+1/RC); Y=X.*H; magY=abs(Y); EE 3512: Lecture 17, Slide 7

Ideal Linear Phase Lowpass Filter Consider the ideal lowpass filter with frequency response: Phase Response B , j dt , e B B = j ( ) H e 0 B Using the Fourier transform pair for a rectangular pulse, and applying the time-shift property: Impulse Response B B = ( ) sin ( h t c t dt Is this filter causal? The frequency response of an ideal bandpass filter can be similarly defined: = elsewhere , 0 j dt , e B B j 1 2 ( ) H e Will this filter be physically realizable? Why? EE 3512: Lecture 17, Slide 8

Nonlinear Amplifier ( ) t 0 = ( ) cos x t A Assume we apply a sinewave input to a voltage amplifier under two conditions: (t ) y Voltage Amplifier vout = Linear: v a v out in = cos( ) vout aA 0t In this case, the output is: vin The output is a scaled version of the input. = = 2) cos( (in v ( A v a a Nonlinear: v out out ) 2 ( ) aA = = = + 2 2 2 2 ( )) ( ) cos ( ) 1 ( cos( 2 )) a v t aA t t 0 0 0 in 2 Hence, the output signal is at a frequency twice that of the input, which clearly makes this a nonlinear system. In practice, amplifiers, such as audio power amplifiers, are characterized using a power series: ... 3 2 1 0 + + + + = in in in out v a v a v a a v 2 3 A single sinewave input generates many frequencies, all harmonically related to the input frequency. This distortion is characterized by figures of merit such as total harmonic distortion and intermodulation. See audio system measurements for more information. EE 3512: Lecture 17, Slide 9

Summary Reviewed basic systems concepts such as linearity and causality. Reviewed temporal solutions using convolution. Reviewed frequency domain solutions using the Fourier Transform. Reviewed the output of an LTI system to a periodic input. Discussed ideal filters. Discussed nonlinearities and how such a system behaves for periodic inputs. EE 3512: Lecture 17, Slide 10