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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 discretetime 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 discretetime 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 discretetime 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'Ll 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 (Sisl = 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 realyaIUed energy (potser] 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 discretetime 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] = Eixxlkl {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 discretetime 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 DiscreteTime 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 inputoutput 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}
 : 130.; = {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 DiscreteTime 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 zeroinput 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 steadystate response. Plot these two responses for n. =
419. Determine the impulse response for the cascade of two linear timeinvariant 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 ﬁniteduration 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 ...
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