Types of Digital Signals Unit step signal u n 1 n 0 0 n 0 Unit impulse unit sample n u n n X 1 n 0 0 n 6 0 m summing m n u n u n 1 differencing Complex exponentials cisoids x n A exp j n obtained by sampling an analog cisoid xa t A exp j t EE 524 Fall 2004 2 1 i e x n xa nT where T is the sampling interval Thus T or equivalently f F Fs using Fs 1 T 2 f 2 F Sinusoids x n A sin n Useful properties exp j n cos n j sin n cos n sin n EE 524 Fall 2004 2 exp j n exp j n 2 exp j n exp j n 2j 2 A sine wave as the projection of a complex phasor onto the imaginary axis EE 524 Fall 2004 2 3 Sampled vs Analog Exponentials Analog exponentials and co sinusoids are periodic with T 2 discrete time sinusoids are not necessarily periodic although their values lie on a periodic envelope Periodicity condition also for sines and cosines x n x n N ej n ej n N exp j N 1 2 m m m integer or f 2 f N N For sampled exponentials the frequency is expressed in radians rather than radians second Digital signals have ambiguity EE 524 Fall 2004 2 4 Ambiguity in Discrete time Signals Ambiguity Condition for Discrete time Sinusoids sin 1T sin 2T 1 6 2 m 2 1 1 2 m F1 F2 mFs m 1 2 T 2 F1T 2 F2T 2 m EE 524 Fall 2004 2 5 Example lowpass signal with spectrum X F 2 concentrated in the interval Fm Fm Taking F1 Fm and F2 Fm it follows that there is no ambiguity if the signal is sampled with Fs 1 2Fm T where Fs is the sampling frequency The above equation is a particular form of the sampling theorem The frequency FN 2Fm is referred to as the Nyquist rate Discrete time signal ambiguity is often termed as the aliasing effect EE 524 Fall 2004 2 6 Discrete time Systems y n T x n where T denotes the transformation operator that maps an input sequence x n into an output sequence y n Linear system a system is linear if it obeys the superposition principle The response of the system to the weighted sum of signals corresponding weighted sum of the responses outputs of the system to each of the individual input signals Mathematically T ax1 n bx2 n aT x1 n bT x2 n ay1 n by2 n ax1 n bx2 n EE 524 Fall 2004 2 Linear system T ay 1 n by2 n 7 Example Square law device Let y n x2 n i e T 2 Then T x1 n x2 n x21 n x22 n 2x1 n x2 n 6 x21 n x22 n Hence the system is nonlinear A time invariant or shift invariant system has input output properties that do not change in time if y n T x n y n k T x n k Linear time invariant LTI system is a system that is both linear and time invariant sometimes referred to as linear shiftinvariant LSI system EE 524 Fall 2004 2 8 Discrete time Signals via Shifted Impulse Functions x n X x k n k k EE 524 Fall 2004 2 9 Response of LTI System Let h n be the response of the system to n Due to the time invariance property the response to n k is simply h n k Thus y n T x n T X x k n k k X x k T n k k X x k h n k k x n h n convolution sum The sequence h n impulse response of LTI system EE 524 Fall 2004 2 10 Convolution Properties An important property of convolution x n h n X x k h n k k X h k x n k k h n x n i e the order in which two sequences are convolved is unimportant Other properties x n h1 n h2 n associativity x n h1 n h2 n x n h1 n h2 n distributivity x n h1 n x n h2 n EE 524 Fall 2004 2 11 Stability of LTI Systems An LTI system is stable if and only if X h k k Proof absolute summability stability Let the input x n be bounded so that x n Mx n Then y n X h k x n k k h k x n k k X Mx X h k y n if k X h k k P Now it remains to prove that if k h k then a bounded input can be found for which the output is not bounded Consider h n h n h n 6 0 x n 0 h n 0 X y 0 k if P k h k EE 524 Fall 2004 2 h k x k X h k k the output sequence is unbounded 12 Causality of LTI Systems Definition A system is causal if the output does not anticipate future values of the input i e if the output at any time depends only on values of the input up to that time Thus a causal system is a system whose output y n depends only on x n 2 x n 1 x n Consequence A system y n T x n is causal if whenever x1 n x2 n for all n n0 then y1 n y2 n for all n n0 where y1 n T x1 n y2 n T x2 n Comments All real time physical systems are causal because time only moves forward Imagine that 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 broadcasts Proposition An LTI system is causal if and only if its impulse response h n 0 for n 0 Proof From the definition of a causal system y n X h k x n k k EE 524 Fall 2004 2 13 X h k x n k k 0 P 1 Obviously this equation is valid if k h k x n k 0 for all x n k h n 0 for n 0 The other direction is obvious 2 If h n 6 0 for n 0 system is noncausal h n 0 n 1 h 1 6 0 y n X h k x n k h 1 x n 1 k 0 y n depends on x n 1 noncausal system EE 524 Fall 2004 2 14 Example An LTI system with n h n a u n an n 0 0 n 0 Since h n 0 for n 0 the system is causal To decide on stability we must compute …