Lecture 02 Discrete-time signals and systems, part 1

1 2. Discrete-time signals AND SYSTEMS, part 1. 1. Lecture 2 - 36 minutes x(O). x(1)x(2) General Seque x(n). nce Graphical representa- tion of discrete - time signals *a ?III 7 8 91011 n -3-2-1 0 1 2 3 4 5 6. Unit Sample(Impulse) b(n). 8(n)=1 n=O. =0 Otherwise -101 - -*-n -1 0 1 2 3. u(n). 00*. 8(n) :u(n) -u(n-1) The unit-sample sequence in terms of the unit- step sequence. 21111 .. u (n). -~~~ - u(n-1). u(n)= Z 8(k) The unit-step sequence in terms of the unit- sample sequence. n<(. | 1 8(k n1. 8(k). Rea IExponential Exponential and x(n)= (n Sinusoidal sequences. O n Sinusoidal x(n)=Acos(wen+O). * * wo= 4=. j 0 j nj x(-1) ()1)x(2) x(n) Representation of an arbitrary sequence as a linear combina- -10 1. (O) I x(0)8(n). n tion of delayed unit samples. -1 1(2 x(O)8(n)+x(1)8(n-1). -- X(1I x(1)8n-l) +x(-1)B(n+1)+--- -1 0 12 n =I x(k)8(n-k).)))

2 K=-OD. +1). xee9oo (-1)8(. n -1 0 1 2. 1-B- 1- n4. x(-2 x(-2) 8(n+2). -1a 1 2. ". y(n)= E x(k)h(n-k) Illustration of k=-O folding and shifting for linear convolu- xxxxxxxxxxx xWk tion. -101234 k N N 0N hk -101234 k N h(O-k). -101 234. 00 H h(-4-k). -1012 34. 2. Correction In the Lecture I indicate that the sinusoidal sequence A cos(w n + #) with w = 3ff/7 and # = - Tr/8 is not periodic. In fact it is peri8 dic although Rot with a period of 2rr/we. (See problem (a)). For w0 = 3/7 the sinusoidal sequence will not be periodic. 3. Comments In this Lecture we introduce the class of Discrete-time signals and systems. The unit sample, unit step, exponential and sinusoidal sequences are basic sequences which play an important role in the analysis and representation of more complex sequences. The class of Discrete-time systems that we focus on is the class of linear shift- invariant systems.)

3 The representation of this class of systems through the convolution sum and some properties of convolution are developed. 4. Reading Text: Section (page 8) through eq. ( ) page 28 section 5. Problems Problem Determine whether or not each of the following sequences is periodic. If your answer is yes, determine the period. (a) x(n) = A cos (- n-). - f). (b) x(n) = e (n/8. Problem A sequence x(n) is shown below. Express x(n) as a linear combination of weighted and delayed unit samples. -4 -3 -2 -1 0. 1 2 3 4. Figure Problem For each of the following systems, y(n) denotes the output and x(n). the input. Determine for each whether the specified input-output relationship is linear and/or shift-invariant. (a) y(n) = 2x(n) + 3. (b) y(n) = x(n) sin(2 n + ). (c) y(n) = (x(n)]2. n (d) y (n) =, xx(m). m=-_O. Problem For each of the following pairs of sequences, x(n) represents the input to an LSI system with unit-sample response h(n).)

4 Determine each output y(n). Sketch your results. (a). x(n) 2. -l 0 1 2. h(n) = u(n). 0 1 2. Figure x(n). (b) 2. -2 -l 0 1 2. h(n). -2. Figure x(n) = an u(n) 0 < a <l (c). I I I I T. h(n) = n u(n) ; 0 < < ; / a 0 0 0. 0. Figure (d). x (n) u (n). * . *. 0 1. h (n) 1. 3 4 5. -l 0 1 2. -1. Figure The following formulas may be useful: C a = , [a| < 1. E 1-a r=0. N-1 r 1-aN. aE 1-a , all a r=0. Problem The system shown below contains two linear shift-invariant subsystems with unit sample responses h1 (n) and h2 (n), in cascade. V(n) .. _ y (n). h (n) = 6(n) - 6 (n - 3). 0. h 2 (n) = (.8)n u(n). L I TI T 1*. 0. Figure (a) Let x(n) = u(n). Find ya (n) by first convolving x(n) with h1 (n) and then convolving the result with h 2 (n) ya(n) = [x(n) * h1 (n)] * h 2 (n). (b) Again let x(n) = u(n). Find yb(n) by convolving x(n) with the result of the convolution of h1 (n) and h 2 (n) yb(n) = x(n) * [h1 (n) * h 2 (n)].]

5 Your results for parts (a) and (b) should be identical, illustrating the associative property of convolution. Problem *. If the output of a system is the input multiplied by a complex constant then that input function is called an eigenfunction of the system . (a) Show that the function x(n) = zn, where z is a complex constant, is an eigenfunction of a linear shift-invariant Discrete-time system . (b) By constructing a counterexample, show that znu(n) is not an eigenfunction of a linear shift-invariant Discrete-time system . * Asterisk indicates optional problem. MIT OpenCourseWare Resource: Digital signal Processing Prof. Alan V. Oppenheim The following may not correspond to a particular course on MIT OpenCourseWare, but has been provided by the author as an individual learning resource. For information about citing these materials or our Terms of Use, visit.