Questions and Answers of Signals and Systems

Pure tones or sinusoids are not very interesting to listen to. Modulation and other techniques are used to generate more interesting sounds. Chirps, which are sinusoids with time-varying frequency,
The input-output equation characterizing an amplifier that saturates once the input reaches certain values iswhere x(t) is the input and y(t) the output.(a) Plot the relation between the input x(t)
The following problems relate to linearity, time-invariance, and causality of systems.(a) A system is represented by the equation z(t) = v(t) f(t) + B where v(t) is the input, z(t) the
A fundamental property of linear time-invariant systems is that when-ever the input of the system is a sinusoid of a certain frequency the output will also be a sinusoid of the same frequency but
The impulse response of an LTI continuous-time system is h(t) = u(t) ˆ’ u(t ˆ’ 1).(a) If the input of this system is x(t), is it true that the system output is(b) If the input is
The input of an LTI continuous-time system with impulse response h(t) = u(t) ˆ’ u(t ˆ’ 1) is(a) Find the output y(t) of the system using the convolution integral.(b) If T = 1,
The voltage-current characterization of a p-n diode is given by (see Figure 2.22)i(t) = Is(eqv(t)/kTˆ’1)where i(t) and v(t) are the current and the voltage in the diode (in the direction
Consider an envelope detector that is used to detect the message sent in the AM system shown in the examples. The envelope detector as a system is composed of two cascaded systems: one which computes
Frequency modulation, or FM, uses a wider bandwidth than amplitude modulation, or AM, but it is not affected as much by noise as AM is. The output of an FM transmitter is of the formwhere m(t) is the
The support of a period of a periodic signal relates inversely to the support of the line spectrum. Consider two periodic signals: x(t) of fundamental period T0 = 2 and y(t) of fundamental period T1
In the computer generation of musical sounds, pure tones need to be windowed to make them more interesting. Windowing mimics the way a musician would approach the generation of a certain sound.
Consider a saw-tooth signal x(t) with fundamental period T0= 2 and period(a) Find the Fourier coefficients Xk using the Laplace transform. Consider the cases when kis odd and even (k
The modulation-based frequency transformation of the DTFT is applicable to IIR filters. It is obvious in the case of FIR filters, but requires a few more steps in the case of IIR filters. In fact, if
A low-pass IIR discrete filter has a transfer function(a) Find the poles and zeros of this filter.(b) Suppose that you multiply the impulse response of the low-pass filter by
A second-order analog Butterworth filter has a transfer function(a) Is the half-power frequency of this filter Î©hp = 1 rad/sec?(b) To obtain a discrete Butterworth filter
An FIR filter has a system function H(z) = 0.05z2 + 0.5z + 1 + 0.5z−1 + 0.05z−2.(a) Find the magnitude |H(ejω)| and phase response ∠H(ejω) at frequencies ω = 0, π/2 and π.
The impulse response h[n] of an FIR is given by h[0] = h[3], h[1] = h[2], the other values are zero.(a) Find the Z-transform of the filter H(z)and the frequency response H(ejω).(b) Let
If x[n] is periodic of period N1> 0 and y[n] is periodic of period N2> 0(a) What should be the condition for the sum z[n] of x[n] and y[n] to be periodic?(b) What would be the period
A discrete-time averager is represented by the input/output equation y[n] = (1/3)(x[n + 1] + x[n] + x[n − 1]), where x[n] is the input and y[n] the output.(a) Determine whether this system is
The following difference equation is used to obtain recursively the ratio α/βc[n + 1] = (1 − β) c [n] + α n ≥ 0with c[0] as an initial
A sinusoid x(t) = cos(t) is a band-limited signal with maximum frequency Ωmax = 1(a) Using Fourier transform properties determine the maximum frequency of x2(t). What sampling period Ts can be
Consider the impulse response of a LTI system h(t) = eˆ’at[u(t) ˆ’ u(t ˆ’ 1)] a >0.(a) Obtain the transfer function H(s).(b) Find the poles and zeros of
The transfer function of a causal LTI system is(a) Find the ordinary differential equation that relates the system input x(t)to the system output y(t).(b) Find the input x(t)so that for
We would like to find the Fourier series of a saw-tooth periodic signal x(t) of period T0 =1. The period of x(t) isx1(t) = r(t) − r(t − 1) − u(t − 1)(a) Sketch x(t) and compute the
The unit-step response of a system is s(t) = [0.5−e−t + 0.5e−2t] u(t).(a) Find the transfer function H(s)of the system.(b) How could you use s(t)to find the impulse response h(t)and the
Consider the following impulse responsesh1(t) = [(2/3)e−2t + (1/3)et] u(t),h2(t) = (2/3)e−2t u(t) − (1/3)et u( − t),h3(t) = −(2/3)e−2t u( − t) − (1/3)et u( − t)(a) From the
Consider a LTI system with transfer function(a) Determine if the system is BIBO stable or not.(b) Let the input be x(t) = cos(2t) u(t) find the response y(t)and the corresponding
A continuous-time periodic signal x(t) with fundamental period T0 = 2 has a period x1(t) = u(t) − u(t − 1).(a) Is x(t)a band-limited signal? Find the Fourier coefficients Xk of
The output of an ideal low-pass filter is(a) Assume the filter input is a periodic signal x[n]. What is its fundamental frequency Ï‰0? What is the fundamental period N0?(b) When
You have designed an IIR low-pass filter with an input-output relation given by the difference equation(i) y[n] = 0.5y[n − 1] + x[n] + x[n − 1] n ≥ 0where
For simple signals it is possible to obtain some information on their DTFTs without computing it. Letx[n] = Î´[n] + 2Î´[n ˆ’ 1] + 3Î´[n
Let x1[n] = 0.5 n ,0 ≤ n ≤ 9 be a period of a periodic signal x[n].Use the Z-transform to compute the Fourier series coefficients of x[n].
Inputs to an ideal low-pass filter with frequency response
The frequency response of a filter is H(ejω) = 1.5 + cos(2ω),− π ≤ ω ≤ π.(a) Is |H(ejω)| = H(ejω)? Is phase zero?(b) Find the impulse response h[n]of this filter. What type of
An FIR filter has a transfer function H(z) = zˆ’2(zˆ’ejÏ€/2)(zˆ’eˆ’jÏ€/2).(a) Find and plot the poles and zeros of this
The impulse response of an FIR filter is h[n] = αδ[n] + βδ[n − 1] + αδ[n − 2], α > 0 and β > 0.(a) Determine the value of α and β for which this filter has a dc gain
The transfer function of an FIR filter is H(z) = z−2 (z − 2)(z − 0.5).(a) Find the impulse response h[n] of this filter and plot it. Comment on any symmetries it might have.(b) Find
The transfer function of an IIR filter is(a) Calculate the impulse response h[n]of the filter.(b) Would it be possible for this filter to have linear phase? Explain.(c) Sketch the
The transfer function of an IIR filter isFind the magnitude response of this filter at Ï‰ = 0, Ï‰ = Ï€/2, and Ï‰ = Ï€.From the poles and the zeros of
Consider the following problems related to the specification of IIR filters(a) The magnitude specifications for a low-pass filter are1 – δ ≤ |H(ejω)| ≤ 1
A first-order low-pass analog filter has a transfer function H(s) = 1/(s + 1).(a) If for this filter, the input is x(t) and the output is y(t) what is the ordinary differential equation
Given the discrete IIR filter realization shown in Figure 12.31 where G is a gain value(a) Determine the difference equation that corresponds to the filter realization.(b) Determine the range
Consider the following transfer function:(a) Develop a cascade realization of H(z) using a first-order and a sec-ond-order sections. Use minimal direct form to realize each of the sections.(b)
Given the realization in Figure 12.32. Obtain(a) the difference equations relating g[n] to x[n]and g[n] to y[n],(b) the transfer function H(z) = Y(z)/X(z) for this filter.Figure 12.32:
A three-point moving-average filter is of the form:y[n] = β(αx [n − 1] + x[n] + αx[n + 1])where α and β are constants, and x[n] is the input and y[n] is the output of the
Let the filter H(z) be the cascade of a causal filter with transfer function G(z) and an anti-causal filter with transfer function G(zˆ’1), so thatH(z) = G(z)
FIR and IIR filters: symmetry of impulse response and linear-phase— Consider two FIR filters with transfer functionsH1(z) = 0.5 + 0.5z−1 + 2.2z−2 + 0.5z−3 + 0.5z−4H2(z) = − 0.5 −
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 Î±1, Î±2, 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] ˆ’ u[n ˆ’ 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] = ejÎ¸Î´[n + Ï„] + eˆ’jÎ¸Î´[n ˆ’ 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] ˆ’ u[n ˆ’ 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] = Î²n u[n], Î² > 0, for what values of Î² 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 ˆ’ 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 Î± > 0. Find values of Î±for 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 ˆ’ 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

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