Questions and Answers of Signals and Systems

When we pad an aperiodic signal with zeros, we are improving its frequency resolution, i.e., the more zeros we attach to the original signal the better the frequency resolution, as we
A definite advantage of the FFT is that it reduces considerably the computation in the convolution sum. Thus if x[n], 0 ‰¤ n ‰¤ N ˆ’ 1, is the input of
Consider the circular convolution of two signals x[n] = n, 0 ≤ n ≤ 3 and y[n] = 1, n = 0, 1, 2 and zero for n = 3.(a) Compute the convolution sum or linear convolution of x[n] and
Consider a filter with a transfer function(a) Determine the magnitude of this filter at Î© = 0, 1, and ˆž. What type of filter is it?(b) Show that the bandwidth of this
A series RC circuit is connected to a voltage source vi(t), and the output is the voltage across the capacitor, vo(t).(a) Find the transfer function H(s) = Vo(s)/Vi(s) of this filter when the
Consider a second-order analog filter with transfer function(a) Determine the dc gain of this filter. Plot the poles and zeros; determine the magnitude response |H(jÎ©)|of this filter
Consider a first-order system with transfer functionwhere K > 0 is a gain, z1 is a zero, and p1 is a pole.(a) If we want unity dc gain, i.e., |H(j0)| = 1, what should be the value of
A second-order analog low-pass filter has a transfer functionwhere Q is called the quality factor of the filter.(a) Show that the maximum of the magnitude€¢ for Q <
A passive RLC filter is represented by the ordinary differential equationwhere x(t) is the input and y(t) is the output.(a) Find the transfer function H(s) of the filter and indicate what type
The receiver of an AM system consists of a band-pass filter, a demodula-tor, and a low-pass filter. The received signal isr(t) = m(t)cos(40000π t) + q(t)where m(t) is a desired voice signal with
Consider the RLC circuit in Figure 7.15 where R = 1Î©.(a) Determine the values of the inductor and the capacitor so that the transfer function of the circuit when the output is
The loss at a frequency Ω = 2000(rad/sec) is α(2000) = 19.4 dBs for a fifth-order lowpass Butter worth filter with unity dc gain. If we let α(Ωp) = αmax = 0.35 dBs, determine• The half-power
The specifica-tions for a low-pass filter areΩp = 1500 rad/sec, αmax = 0.5 dBsΩs = 3500 rad/sec, αmin = 30
Consider the following low-pass filter specificationsαmax = 0.1 dB αmin = 60 dBΩp = 1000 rad/sec Ωs = 2000 rad/sec(a) Use MATLAB to design a Chebyshev low-pass filter that satisfies the
A desirable signal x(t) = cos(100πt) – 2 cos(50π t) is recorded as y(t) = x(t) + cos(120πr t), i.e., as the desired signal but with a 60 Hz hum. We would like to get rid of the hum and recover
Consider the sampling of real signals.(a) Typically, a speech signal that can be understood over a telephone line shows frequencies from about 100 Hz to about 5 kHz. What would be the sampling
Consider the sampling of a sinc signal and related signals.(a) For the signal x(t)=sin(t)/t, find its magnitude spectrum |X(Ω)|and determine if this signal is band-limited or not.(b) What would be
Consider the signal x(t)=2 sin(0.5t)/t(a) Is x(t) band-limited? If so, indicate its maximum frequency Ωmax.(b) Suppose that Ts = 2 π, how does Ωs relate to the Nyquist frequency 2Ωmax?
Consider the signal x(t) = δ(t + 1) + δ(t − 1).(a) Find its Fourier transform X(Ω). Determine if x(t) is band-limited or not. If band-limited, give its maximum frequency.(b) Filtering x(t) with
The signal x(t) has a Fourier transform X(Ω) = u(Ω + 1) − u(Ω − 1) thus it is band-limited, suppose we generate a new signal y(t) = (x* x)(t), i.e., it is the convolution of x(t)
Suppose you wish to sample an amplitude modulated signal x(t) = m(t) cos(Ωct) where m(t) is the message signal and Ωc = 2π104(rad/sec) is the carrier frequency.(a) If the message is an acoustic
The input/output relation of a non-linear system is y(t) = x2(t), where x(t) is the input and y(t) is the output.(a) The signal x(t) is band-limited with a maximum frequency ΩM = 2000π(rad/sec),
A message m(t) with a bandwidth of B = 2 kHz modulates a cosine carrier of frequency 10 kHz to obtain a modulated signal s(t) = m(t)cos(20 × 103πt).(a) What is the maximum frequency of s(t)? What
You wish to recover the original analog signal x(t) from its sampled form x(nTs).(a) If the sampling period is chosen to be Ts = 1 so that the Nyquist sampling rate condition is satisfied, determine
A periodic signal has the following Fourier series representation(a) Is x(t) band-limited?(b) Calculate the power Px of x(t).(c) If we approximate x(t) asby using 2N + 1 terms, find N so that
Suppose x(t) has a Fourier transform X(Ω) = u(Ω + 1) − u(Ω − 1)(a) Determine possible values of the sampling frequency Ωs so that x(t) is sampled without
Consider the periodic signalsx1(t) = cos(2π t), x2(t) = cos((2π + φ)t)(a) Let φ = 4π, show that if we sample these signals using Ts =
A signal x(t) is sampled with no aliasing using an ideal sampler. The spectrum of the sampled signal is shown in Figure 8.17.(a) Determine the sampling period Ts used.(b) Determine the
Consider the signals x(t) = u(t) − u(t − 1), and y(t) = r(t) − 2r(t − 1) + r(t − 2)(a) Is any of these signals band-limited? Explain.(b) Use Parseval’s energy result to
Signals of finite time support have infinite support in the frequency domain, and a band-limited signal has infinite time support. A signal cannot have finite support in both
Suppose you want to find a reasonable sampling period Ts for the non-causal exponential x(t)= e−|t|.(a) Find the Fourier transform of x(t), and plot |X(Ω)|. Is x(t) band-limited? Use the
Let ‰¤ t ‰¤ 1 ‰¤ t ‰¤ 1, and zero otherwise, be the input to a 2 bit analog-to-digital converter.(a) For a sampling period Ts = 0.025 sec.
For the discrete-time signalsketch and label carefully the following signals:(a) x[n ˆ’ 1], x[ˆ’n], and x[2 ˆ’ n].(b) The even component xe[n] of
For the discrete-time periodic signal x[n] = cos (0.7πn),(a) Determine its fundamental period N0.(b) Suppose we sample the continuous-time signal x(t) = cos (πt) with a sampling period Ts
Consider the following problems related to the periodicity of discrete-time signals.(a) Determine whether the following discrete-time sinusoids are periodic or not. If periodic, determine its
The following problems relate to periodicity and power of discrete-time signals.(a) Is the signal x[n] = ej(n−8)/8 periodic? if so determine its fundamental period N0. What if x1[n] =
The following problems relate to linearity, time-invariance, causality, and stability of discrete-time systems.(a) The output y[n] of a system is related to its input x[n] by y[n] = x[n]x[n
Consider a discrete-time system with output y[n] given by y[n] = x[n] f[n] and x[n] is the input and f[n] is a function.(a) Let the input be x[n] = 4cos (πn/2) and f[n] = cos (6πn/7), −∞
Consider a system represented bywhere the input is x[n] and the output y[n]. Is the system(a) linear? time-invariant?(b) causal? bounded-input bounded-output stable? n+4 γin] Σt x[k] k=n-2
You are testing a 1 volt. d.c. source and have the following measurements obtained from the source every minute starting at time 0To find the average voltages for the first 5 min, i.e., to get rid of
A continuous-time system is characterized by the ordinary differential equationThis equation is discretized by approximating the derivatives for a signal Ï(t) asaround t = nT, and for a
A causal, LTI discrete-time system is represented by the block diagram shown in Figure 9.15 where D stands for a one-sample delay.(a) Find the difference equation relating the input x[n] and the
The input and the output of an LTI causal discrete-time system areInput: x[n] = u[n] − u[n − 3], Output : y[n] = u[n − 1] − u[n − 4](a) What should be the
The following problems relate to the response of LTI discrete-time systems.(a) The unit-step response of a LTI discrete-time system is found to be s[n] = (3 − 3(0.5)n+1 )u[n]. Use s[n] to
An LTI discrete-time system has an impulse response h[n] = u[n] − u[n − 4], and as input the signal x[n] = u[n] − u[n − (N + 1)] for a positive integer N. The output of the system y[n]
Consider a discrete-time system represented by the difference equa-tion y[n] = 0.5y[n ˆ’ 1] + x[n] where x[n] is the input and y[n] the output.(a) An equivalent representation of the
An LTI discrete time system has the impulse response h[n] = (−1)n u[n]. Use the convolution sum to compute the output response y[n],n ≥ 0, when the input is x[n] = u[n] − u[n − 3] and the
An LTI causal discrete-time system has the input/output relationshipwhere x[n] is the input of the system, y[n] is the response of the system.There is zero initial energy in the system prior to
A discrete-time averager is characterized by the following equation relating the input x(nTs) with the output y(nTs)(a) Is this system causal? Explain.(b) Let N = 2 in the above equation.
Consider a causal LTI system with impulse response h[n], and input x[n] = x1[n] ˆ’ x1[n ˆ’ 2] + x1[n ˆ’ 4] where x1[n] = u[n]ˆ’ u[n ˆ’ 2]. The
An LTI system represented by the difference equation y[n] = ˆ’ y[n ˆ’ 1] + x[n], n ‰¥ 0, is initially at rest. The input of the system is x[n] and the output
The output of a discrete-time system is y[n] = w[n] x[n] where x[n] is the input, and w[n] = u[n] − u[n − 5] is a rectangular window.(a) The input is x[n] = 4 sin (πn/2), −∞ < n <
A finite impulse response (FIR) filter has an input/output relation y[n] = x[n] − x[n − 5] where x[n] is the input and y[n] the output.(a) Find the impulse response h[n] of this filter. Plot
Consider the following problems related to properties of filters.(a) Filters that operate under real-time conditions need to be causal, i.e., they can only process present and past inputs. When
Consider the formulax[n] = x[n − 1] + x[n − 3] n ≥ 3x[0] = 0x[1] = 1x[2] = 2Find the rest of the sequence for 0 ≤ n ≤ 50 and plot it
Given the discrete signal x[n] = 0.5nu[n].(a) Use the function stem to plot the signal x[n] for n = ˆ’5 to 20.(b) Is this a finite-energy discrete-time signal? i.e., compute the
Consider an analog periodic sinusoid x(t) = cos (3πt + π/4) being sampled using a sampling period Ts to obtain the discrete-time signal x[n] = x(t)|t=nTs = cos(3πTs n + π/4).(a) Determine
Suppose you sample the analog signalwith a sampling period Ts = 0.25 to generate x[n] = x(t)|t=nTs.(a) Use the function stem to plot x[n] and x[ˆ’n] for an appropriate
Periodic signals can be generated by obtaining a period and adding shifted versions of this period. Suppose we wish to generate a train of triangular pulses. A period of the signal is x[n] = 0.5(r[n]
Consider the discrete-time signal x[n] = cos (2πn/7).(a) The discrete-time signal can be compressed by getting rid of some of its samples (down-sampling). Consider the down-sampling by 2. Write
In the generation of music by computer, the process of modulation is extremely important. When playing an instrument, the player typically does it in three stages: (1) rise time, (2) sus-tained time
An A/D converter can be thought of composed of three subsystems: a sampler, a quantizer, and a coder.(a) The sampler, as a system, has as input an analog signal x(t) and as output a
A window is a signal w[n] that is used to highlight part of another signal. The windowing process consists in multiplying an input signal x[n] by the window signal w[n], so that the output is y[n] =
A discrete-time IIR system is represented by the following difference equation y[n] = 0.15y[n−2]+x[n], n ≥ 0 where x[n] is the input and y[n] is the output.(a) To find the impulse response
An FIR filter has a non-recursive input/output relation(a) Find the impulse response h[n] of this filter.Is this a causal and stable filter?Explain(b) Find the unit-step response s[n] for
The impulse response of a discrete-time system is h[n] = ( − 0.5)n u[n].(a) If the input of the system is x[n] = δ[n] + δ[n − 1] + δ[n − 2], use the linearity and time-invariance of the
Suppose an IIR system is represented by a difference equation y[n] = a y[n − 1] + x[n], where x[n] is the input and y[n] is the output.(a) If the input is x[n] = u[n] and it is known that the
The unit-step response of a discrete-time LTI system iss[n] = 2[( − 0.5)n − 1] u[n]Use this information to find(a) The impulse response h[n] of the discrete-time LTI system.(b) The response
The poles of the Laplace transform X(s) of an analog signal x(t) are p1,2 =−1 ± j1, p3 = 0, p4,5 = ±j1, and there are no zeros. If we use the transformation z = esTs with Ts =
The sign functionextracts the sign of a real valued signal, i.e.,(a) Let s[n] = s1[n] + s2[n], x[n] = n, where s1[n] is causal and s2[n] anti-causal; find their Z-transforms and
Given the anti-causal signal x[n]= −αn u[−n](a) Determine its Z-transform X(z), and carefully plot the ROC when α = 0.5 and α = 2. For which of the two values of α does X(ejω) exist?(b)
An analog pulse x(t) = u(t) ˆ’ u(t ˆ’ 1) is sampled using a sampling period Ts= 0.1.(a) Obtain the discrete-time signal x(nTs) = x(t)|t=nTs and plot it as a function of
Consider the signal x[n] = 0.5(1+ [−1]n) u[n](a) Plot x[n]and use the sum definition of the Z-transform to obtain its Z-transform, X(z).(b) Use the linearity property and the Z-transforms
A LTI system is represented by the first-order difference equationy[n] = x[n] − 0.5y[n − 1] n ≥ 0where y[n] is the output and x[n]
When finding the inverse Z-transform of a function with zˆ’1terms in the numerator, zˆ’1can be thought of a delay operator to simplify the calculation. For(a) Use the
A second-order system is represented by the difference equation y[n] = 0.25y[n − 2] + x[n] where x[n] is the input and y[n] the output.(a) For the zero-input case, i.e., when x[n] = 0, find
Consider the following problems related to LTI systems.(a) The impulse response of an FIR filter is h[n]= Î±n(u[n] ˆ’ u[n ˆ’ M])i. Is it true that the transfer
The transfer function of a causal LTI discrete-time system is H(z) = (1 + z−1)/(1 − .5z−1).(a) Find the poles and zeros of H(z). Choose the correct region of con-vergence corresponding to
Suppose we cascade a differentiator and a smoother. The equations for the differentiator is w[n] = x[n] ˆ’ x[n ˆ’ 1] where w[n] is the output and x[n] the input, and for the
An LTI discrete-time system is characterized by the difference equationy[n] + ay[n − 1] + by[n − 2] = x[n]Determine for which of the the following sets of coefficients the system is BIBO
Consider a discrete-time LTI system represented by the difference equation with the given initial conditiony[n] + 0.5y[n − 1] = 2(x[n] − x[n − 1]) n ≥ 0,
Determine the impulse response h[n]of the feedback system shown in Figure 10.18. Determine if the system is BIBO stable.Figure 10.18: e[n] r[n]- Delay y[n]
Consider the following problems for LTI discrete-time systems.(a) The input and the output of a LTI discrete-time system areFind the transfer function H(z).(b) The transfer function of an
The following problems relate to FIR and IIR systems.(a) The input and the output of a causal LTI discrete-time system areDetermine the impulse response h[n].(b) The transfer function H(z)of an
The Z-transform of the unit-step response of a causal LTI discrete-time system isDetermine the impulse response of the system. 1.5 S(z) = 1 1– 0.5z- -1
The impulse response of a causal LTI discrete-time system is(a) If the input of the system is a pulse x[n] = u[n] ˆ’ u[n ˆ’ 3], determine the length of the output of the
The transfer function of an RLC circuit is H(s) = Y(s)/X(s) = 2s/(s2 +2s+1).(a) Obtain the ordinary differential equation with input x(t) and output y(t). Approximating the derivatives by
We are given a noisy signalx(t) = s(t) + Î·(t)where s(t) is the desired signal and Î·(t) is additive noise. From experience, we know that the average power of the desired
The transfer function of a discrete-time system iswith Î± = r1 ejÎ¸1 and Î² = r2ejÎ¸2, where ri > 0 and Î¸i are angles between 0 and
Suppose we cascade a €œdifferentiator€ and a €œsmoother€ systems characterized by the following input/output equationswhere the output of the differentiator
A model for echo generation is shown in Figure 10.20.(a) Calculate the transfer function H(z) = Y(z)/X(z) of the echo sys-tem shown above.(b) Suppose you would like to recover the original
Suppose we are given a finite-length sequence h[n](it could be part of an infinite-length impulse response from a discrete system that has been windowed) and would like to obtain a
The following are matrices for the state variable and the output equations of a LTI systemAssume vi[n], i = 1, 2, are the state variables, and x[n] the input and y[n] the output. Use the
Given the matrices corresponding to the state and output equations for a system with input x[n] and output y[n]:(a) Find the transfer function H(z) = Y(z)/X(z) corresponding to the state
Consider the following two state-variable representationswhere the first is the controller form and the second the observer form. Find the corresponding functions Hc(z) = Yc(z)/Xc(z) and Ho(z) =
Find an invertable transformation represented by the matrixthat changes the controller form into the observer form given in the previous problem. t2 t1 т t4 t3 ||
Find a state variable and output matrix equations corresponding to the transfer function 0.8z – 0.2z z - z+ 0.5 Н(2). ||
Consider the Fibonacci sequence generated by the difference equation f[n] = f[n − 1] + f[n − 2], n ≥ 0 with initial conditions f[ − 1] = 1, f[ − 2] = −1.(a) Find the
Use symbolic MATLAB to find the inverse Z-transform ofand determine x[n] as n †’ ˆž. 2 – z -1 X (z) 2(1+0.25z-)(1+0.5z¬)
Consider a second-order discrete-time system represented by the following difference equation:y[n] − 2r cos(ω0) y[n − 1] + r2y[n − 2] = x[n] n ≥

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