DSP_HW1 - 33 ('haptcr I 1.3 1.4 1.6 1.? Introduction...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 33 ('haptcr I 1.3 1.4 1.6 1.? Introduction Determine whether or not each of the following signals is periodic. In case a signal is periodic. specify its fundamental period. (a) Trutt) : 3cost5r + n/ol (h) stat : 3cost5n + :r/h] (c) .rtnt = 2 explftaflo — at] (d) .rtnt : costnfh’) costrrn/S} (e) J'UI} = costrra/Zt — sintrrnjh'l + 3cos{:ra/4 + :rf3} (a) Show that the fundamental period NP of the signals ,llfltr- -\' _t',t.ttt]=t i‘:[l.l.2.... is given hf.s NI. : N/GCDtir. Ni. o-‘here GCD is the greatest Common divisor of k and N. (b) What is the fundamental period of this set for N = 7‘? {c} What is it for N = if)? lConsider the following analog sinusoidal signal: Nair} = 3sth llitlmt (3} Sketch the signal .rutrt for [l 5 r 5 3t} ins. (b) The signal .rut'rl is sampled with a satrtpling rate F. = 300 samplesls. Determine the frequency of the discrete-time signal rt”) 2 .t'atHTt. T = UK. and show that it is periodic. (Cl Compute the sample values in one period of .rtnl. Sketch .rtnt on the same diagram with .rntr}. What is the period ol'the discrete-time signal in millisecrmds'.1 [d] Can you find a Sampling rate F, such that the signal .rtnl reaches its peak value of3'? What is the minimum F. suitable for this task? A continuotIs—time sinusoid .ratt't with fundamental period 1'}, 2 IN}: is sampled at a rate F. : UT to produce a discrete-time siHUsoid .rtnt = .rfltn'f't. to) Show that .rtnt is periodic if T/ i], : UN {i.e.. T/ If}, is a rational number). (b) If .t'ln} is periodic. what is its fundamental period Tr, in seconds'.l (c) Explain the statement: .rta} is periodic if its fundamental period Ty. in seconds. is equal to an integer number of periods of .rfitrt. An analog signal contains frequencies up to it] kHz. {a} What range of sampling frequencies allows exact reconstruction of this signal from its samples? (bl Suppose that we sample this signal with a sampling frequency 1", = 8 kHz. Examine what happens to the frequency 1"] = 5 kHz. (c) Repeat part (b) for a frequency F: = 1 kHz. 130 {"haptcr2 Discrete—Time Signals and Systems .1'! Ill Figure P2.2 (a) .t'tn — 2) {h} .t'L-l wal {011m + 2] {d} .i‘tnlut2 — n} (e) Yrtn — Into: — 3] {f} Ital} {g} even part ofs'la} (I1) odd part ofxtnl 2.3 Show that {a} :‘Stul =m‘ul — atn —— Il [b] utnl :. Ellhx (Sis-l = Zinéta — it] 2.4 Show that any signal can be decomposed into an even and an odd component. is the decomposition unique? lllustrate your arguments using the signal .rtal = {2.3.-’l.5.h} 2.5 Show that the energy (power) ota real-yaIUed energy (pots-er] signal is equal to the sum of the energies (powers) ol‘ its even and odd components. 2.6 Consider the system yin} = TLrtnl] = start {5:} Determine if the system is time invariant. (b) To clarify the result in part (a) assume that the signal .. U. elsewhere l. t} «:H ‘1 'i Ital: ‘ ‘ is applied into the system. [1} Sketch the signal .t‘tnl. (2) Determine and sketch the signal yin) 2 Thou]. {3) Sketch the signal yitn: = ytn — 2}. {-4) Determine and sketch the signal nth) 2 HM —- 2]. (5} Determine and sketch the signal )3th = Tlsgtnl]. (6) Compare the signals ygtn} and yin — 2}. What is your conclusion'.l (c) Repeal part (b) for the system yin] = .t'iitl — .t‘ta — 1) Can you use this result to make any statement about the time invariance olthis system:l Why? (d) Repeat parts (h) and {c) for the system ytnl = Thom] = aria) 2.7 A discrete-time system can he ('ltaplerE Problems 131 (I) Static or dynamic {'2} Linear or nonlinear (3) Time invariant or time varying (4) Cansal or noncausal {5) Stable or unstable Examine the following systems with respect to the properties above. (a) yth 2 coslxtnl] {I1} yin] = Eixx-lkl {e} ytnt = .rtnlcostmunt (d) yin] = .rt—n + El (e) yttrt = Trun[.rtntl. where Trunla'tnl] denotes the integer part of flat. obtained by truncation {f} ytn] = Roundirtnfl. where RoundLrtnt] denotes the integer part of .rtnl ob- tained by rounding Remark.- The systems in parts (e) and (l'} are quantizers that perform truncation and rounding. respectively. (g) yin) = |.rtnl| (h) yin} =.rtntutnl {i} ytnl = .t'tn} +n_t'tn + It (i) yin} =.t't2nl .t'lnl. illn'lni 3 ll ll. ifs‘tnl <: t] (I) .t'lttl = .t‘l—nl lm) rte) = signlxtnt] (It) Yt'lltl (n) The ideal sampling system with input .t'fltr} and output .rtn] = ,rfltnTt. —:~c < a e 3c 2.8 Two discrete-time systems T. and 5'”; are connected in cascade to form a new system ’1" as shown in Fig. P23. Prove or disprove the following statements. {a} If T. and '1"; are linear. then it" is linear [i.e.. the cascade connection oflwo linear systems is linear). (b) If ’3] and T3 are time invariant. then T is time invariant. (e) If it". and 5‘; are causal. then T is causal. (d) If T1 and '3"; are linear and time invariant. the same holds for T. (e) If S". and '3”; are linear and time invariant. then interchanging their order does not change the system '3". (f) As in part {e} except that if] . T3 are now time varying. (Hint: Use an example.) (g) [1' T1 and T3 are nonlinear. then '5" is nonlinear. (h) If S". and 5'"; are stable. then 3" is stable. (i) Show by an example that the inverses of parts (c) and (it) do not hold in general. 132 Chapter 2 Discrete-Time Signals and Systems Figure P2.B T: TIT. 2.9 Let T be an L'l‘l. relaxed. and 31130 stable system with input .t'tni and output yin). Show that: (a) If It“) is periodic with period N [i.e.. It”) 2 .t'ii'i + N) for all n :_> U]. the output yin} tends to a periodic signal with the same period. (b) if .rtul is bounded and tends to a constant. the output will also tend to a constant. (e) if .t'il'l'} is an energy signal. the output yin} will also be an energy signal. 2.10 “the following input-output pairs have been observed during the operation of a time- invariant system: mm = {1.0.2} {5—) yltnl = {t}. 1.2} I i .t'gil'l'] = {0. 0. 3} L» _i'3t0] = {0. l. 0. 2} | : 1-30.; = {0. 0. 0. I} (—3 ygtnl = {1. 2. t} i i (fan you draw any conclusions regarding the linearity of the system. What is the impulse response of the system”? 2.11 The following input—output pairs have been observed during the operation ofa linear system: mm = {4.3. t} L mm = {La —1.0. H T .t‘giil} = {1. —1ii—J-_t‘3[.fi)=l—i.!.(}.2i mm = {0. t. I} one = n.2, I} Can you draw any conclusions about the time invariance of this system? 2.12 The only availabie information about a system consists of N input—output pairs. of signals _t‘,-tn} : T|.t',-t{ill. i = l. 2. . . . . N. (a) What is the class of input signals for which we can determine the output. using the information above. if the system is known to be linear?' {b} The same as above. if the system is known to be time invariant. 2.13 Show that the necessary and sufficient condition fora relaxed LTI system to be 8130 stable is x. 2 [Hull 3 Ms <: Ii="'\.' for some constant M... 135 Chapter 2 Discrete-Time Signals and Systems 2.26 2.27 2.28 2.29 2.30 2.31 2.32 (a) Show that any sequence 1(a) can be decomposed as rte) = Z t'tytn — it} il'Z—X and express :1. in terms of .rtni. {b} Use the properties of linearity and time invariance to express the output yin} = Tll‘iitll in terms ofthe input .rtii] and the signal gin) : Throat]. where is an LTI system. (c} Express the impulse response htnl : THUG] in terms of gin). Determine the zero-input response of the system described by the second—order difference equation stir} — Erin — li —4_riii —2i = 0 Determine the particular solution of the difference equation 5 l yin} = gym — l) — gym — 2) +102) when the forcing function is .rtn) = Tutu}. In Example 2.4.8. equation (2.4.30). separate the output sequence yin] into the transient response and the steady-state response. Plot these two responses for n. = 419. Determine the impulse response for the cascade of two linear time-invariant systems having impulse responses. mm} : n”[u[n) — “(n — Ni] and right) = {HIM} '— utn — Ml] Determine the response ytni. n 3 ll. of the system described by the second—order difference equation ytn} — 3y“: — l] — éytii — 2} : .t‘tii) + Zrtn ~— ll to the input .rtn) : 4”n(n]. Determine the impulse response of the following causal system: _r(ii} — 3ytn — l) H 4_r(n — 2] 2.11:”) + 2"“! — I] Let rte). NI 5 n 5 N3 and hi"). M1 5 H 3 M3 be two finite-duration signals. (a) Determine the range L. 5 ii 5 L3 of their convolution. in terms of NI. N2. M1 and M]. (b) Determine the limits ofthe cases of partial overlap from the left. full overlap. and partial overlap from the right. For convenience. assume that hm] has shorter duration than .rtn}. (e) illustrate the validity of your results by computing the convolution of the signals 1. —25n54 H") = l (l. elsewhere 2. —15n52 mm = it). elsewhere ...
View Full Document

Page1 / 5

DSP_HW1 - 33 ('haptcr I 1.3 1.4 1.6 1.? Introduction...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online