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signals and systems
Questions and Answers of
Signals and Systems
Consider two filters with transfer functions(a) The magnitude response of these two filters is unity, but that they have different phases. Find analytically the phase of H1(ejÏ) and
A Butterworth low-pass discrete filter of order N has been designed to satisfy the following specifications:Sampling period Ts =100 µ secαmax = 0.7 dB for 0 ≤ f ≤ fp = 1000 Hzαmin = 10 dB for
Bilinear transformation and pole location—Find the poles of the discrete filter obtained by applying the bilinear transformation with K = 1 to frequency normalized analog second-order
Warping effect of the bilinear transformation—The non-linear relation between the discrete frequency ω(rad) and the continuous frequency (rad/sec) in the bilinear transformation causes warping
The warping effect of the bilinear transformation also affects the phase of the transformed filter. Consider a filter with transfer function G(s) = e−5s.(a) Find the transformed discrete
Design a Butterworth low-pass discrete filter that satisfies the following specifications:0 ≤ α(ejω) ≤ 3 dB for 0 ≤ f ≤ 25 Hzα(ejω) ≥ 38 dB for 50 ≤ f ≤ Fs/2 Hzand the sampling
Consider an all-pass analog filter(a) Use MATLAB functions to plot the magnitude and phase responses of G(s). Indicate whether the phase is linear.(b) A discrete filter H(z) is obtained from
We wish to design a discrete Butterworth filter that can be used in filtering a continuous-time signal. The frequency components of interest in this signal are between 0and 1 kHz, so we would like
Let z = 8 + j3 and v = 9 − j2,(a) Find (i) Re(z) + Im(v),
If we wish to preserve low frequencies components of the input, a low-pass Butterworth filter could perform better than a Chebyshev filter. MATLAB provides a second Chebyshev filter function cheby2
The gain specifications of a filter are− 0.1 ≤ 20 log10|H(ejω)| ≤ 0(dB) 0 ≤ ω ≤ 0.2 π20 log10|H(ejω)| ≤ − 60(dB) 0.3π≤ω≤π(a) Find
Notch filters is a family of filters that includes the all-pass filter. For the filter(a) Determine the values of α1, α2, and K that would make H(z) an all-pass
Consider a filter with transfer function(a) Find the gain K so that this filter has unit dc gain. Use MATLAB to find and plot the magnitude response of H(z), and its poles and zeros. Why is it
Consider down-sampling the impulse response h[n] of a filter with transfer function H(z) = 1/(1 − 0.5z−1).(a) Use MATLAB to plot h[n] and the down-sampled impulse response g[n] =
Consider a moving average, low-pass, FIR filter(a) Use the modulation property to convert the given filter into a high-pass filter.(b) Use MATLAB to plot the magnitude responses of the low-pass
Use MATLAB to design a Butterworth second-order low-pass discrete filter H(Z) with half-power frequency θhp = π/2, and dc gain of 1. Consider this low-pass filter a prototype that can be used to
Use MATLAB to design a Butterworth second-order low-pass discrete filter with half-power frequency θhp = π/2, and dc gain of 1, call it H(z). Use this filter as prototype to obtain a filter
From the direct and the inverse DTFT of x[n] = 0.5|n|:(a) Determine the sum
Consider the connection between the DTFT and the Z-transform in the following problems.(a) Let x[n] = u[n + 2] u[n 3].i. Can you find the DTFT X(ejÏ)
A triangular pulse is given byFind a sinusoidal expression for the DTFT of t[n]
The frequency response of an ideal low-pass filter is
Find the DTFT of x[n] = ejθδ[n + Ï] + ejθδ[n Ï], and use it to find the DTFT of
Consider the application of the DTFT properties to filters.(a) Let h[n]be the impulse response of an ideal low-pass filter with frequency response
Find the DTFT X(ejω) of x[n] = δ[n] − δ[n − 2].(a) Sketch and label carefully the magnitude spectrum |X(ejω)| for 0 ≤ ω < 2π.(b) Sketch and label carefully the magnitude
Let x[n] = u[n + 2] u[n 3](a) Find the DTFT X(ejÏ) of x[n] and sketch |X(ejÏ)| vs Ï giving its value at Ï = ±
Consider a LTI discrete-time system with input x[n] and output y[n]. It is known that the impulse response of the system is(a) Determine the magnitude and phase responses
Consider the following problems related to the properties of the DTFT.(a) For the signal x[n] = βn u[n], β > 0, for what values of β you are able to find
The impulse response of an FIR filter is h[n] = (1/3) (δ[n] + δ[n − 1] +δ[n − 2]).(a) Find the frequency response H(ejω), and determine the magnitude and the phase responses for
The transfer function of an FIR filter is H(z) = z −2(0.5z + 1.2 + 0.5z−1).(a) Find the frequency response H(ejω) of this filter. Is the phase response of this filter linear?(b) Find the
Determine the Fourier series coefficients Xi[k], i = 1,..., 4, for each of the following periodic discrete-time signals. Explain the connec-tion between these coefficients and the symmetry of
Determine the Fourier series coefficients of the following periodic discrete-time signals(a) x1 [n] = 1 − cos(2πn/3), x2[n] = 2 + cos(8πn/3), x3[n] = 3 − cos(2πn/3) + cos(8πn/3), x4 [n]
A periodic discrete-time signal x[n] with a fundamental period N = 3 is passed through a filter with impulse response h[n] = (1/3) (u[n] − u[n − 3]). Let y[n] be the filter output. We begin
For the periodic discrete-time signal x[n] with a period x1[n] = n, 0 ≤ n ≤ 3 use its circular representation to findx[n − 2], x[n + 2], x[ − n], x[ − n + k], for
Let x[n] = 1+ejω0n and y[n] = 1 + ej2ω0 n be periodic signals of fundamental period ω0 = 2π/N, find the Fourier series of their product z[n] = x[n] y[n] by(a) calculating the product x[n]
The periodic signal x[n] has a fundamental period N0 = 4, and a period is given by x1[n] = u[n] − u[n − 2]. Calculate the periodic convolution of length N0 = 4 of(a) x[n] with
Consider the aperiodic signalFind the DFT of length L = 4 of(i) x[n], (ii) x1[n] = x[n 3],
Consider the discrete-time signal x[n] = u[n] − u[n − M] where M is a positive integer.(a) Let M = 1, calculate and sample the DTFT X(ejω) in the frequency domain using a sampling frequency
The convolution sum of a finite sequence x[n] with the impulse response h[n] of an FIR system can be written in a matrix form y = Hx where H is a matrix, x and y are input and output values. Let h[n]
The signal x[n] = 0.5n (u[n] − u[n − 3]) is the input of a LTI system with an impulse response h[n] = (1/3) (δ[n] + δ[n − 1] + δ[n − 2]).(a) Determine the length of the output
The input of a discrete-time system is x[n] = u[n] − u[n − 4] and the impulse of the system is h[n] = δ[n] + δ[n − 1] + δ[n − 2](a) Calculate the DFTs of h[n], x[n] of length N
Given the impulse responsewhere α > 0. Find values of αfor which the filter has zero phase. Verifiy your results with MATLAB. -2
An IIR filter is characterized by the following difference equation y[n] = 0.5y[n − 1] + x[n] − 2x[n − 1], n ≥ 0, where x[n] is the input and y[n] the output of the filter.
Consider a moving average FIR filter with an impulse responseLet H(z) be the Z-transform of h[n].(a) Find the frequency response H(ejÏ) of the FIR filter.(b) Let the impulse response
When designing discrete filters the specifications can be given in the time domain. One can think of converting the frequency domain specifications into the time domain. Assume you
Consider the pulses x1[n] = u[n] − u[n − 20] and x2[n] = u[n] − u[n − 10], and their product x[n] = x1[n] x2[n].(a) Plot the three pulses. Could you say that x[n] is a down-sampled
Suppose you cascade an interpolator (an upsampler and a low-pass filter) and a decimator (a low-pass filter and a downsampler).(a) If both the interpolator and the decimator have the same
Let X(ejω) = 2e−j4ω, − π ≤ ω < π.(a) Use the MATLAB functions freqzand angle to compute the phase of X(ejω) and then plot it. Does the phase computed by MATLAB appear
A window w[n] is used to consider the part of a signal we are interested in.(a) Let w[n] = u[n] − u[n − 20] be a rectangular window of length 20. Let x[n] = sin(0.1πn) and we are
The Fourier series of a signal x[n] and its coefficients Xkare both periodic of the same value Nand as such can be written(a) To find the x[n], 0 ¤ n ¤ N
A periodic signal x[n] of fundamental period N can be represented by its Fourier seriesIf you consider this a representation of x[n](a) Is x1 [n] = x[n N0] for any value of N0
Let x[n] be an even signal, and y[n] an odd signal.(a) Determine whether the Fourier coefficients Xk and Yk corresponding to x[n] and y[n] are complex, real, or imaginary.(b) Consider
Suppose you get noisy measurementsy[n] = ( − 1)n x [n] + Aη[n]where x[n] is the desired signal, and η[n] is a noise that varies from 0 to 1 at random.(a) Let A = 0, and x[n] =
Consider a signal x[n] = 0.5n(0.8)n (u[n] − u[n − 40])(a) To compute the DFT of x[n] we pad it with zeros so as to obtain a signal with length 2γ, larger than the length of x[n] but the
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 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 Ω = 0, 1, and . 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(jΩ)|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 = 1Ω.(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 1], x[n], and x[2 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 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
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