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computer science
signals and systems
Questions and Answers of
Signals and Systems
The Fourier transforms of even and odd functions are very important. Let x(t) = e??t? and y(t) = e?t u(t) ? et u(?t). (a) Plot x(t) and y(t), and determine whether they are even or odd. (b) Show that
To understand the Fourier series consider a more general problem, where a periodic signal x(t), of period T0, is approximated as a finite sum of terms where {?k (t)} are ortho-normal functions. To
As you know, π is an irrational number that can only be approximated by a number with a finite number of decimals. How to compute this value recursively is a problem of theoretical interest. In this
The problem with thresholding the DCT coefficients of an image to compress it, is that the locations of the chosen coefficients are arbitrary and difficult to code. Consider then using a mask W(k,
An image can be blurred by means of a Gaussian filter which has an impulse response(a) If h[m, n] = h1[m]h1[n], i.e., separable determine h1[n]. Find the DFT of h[m, n] and plot its magnitude
Image filtering using 2D-FFT — Consider the linear filtering of an image using the 2D-FFT. Load the image clown and use three different filters to process it given in different formats.• Low-pass
To compute the 2D-DFT one can use 1D-DFT by separating the equation for the 2D-DFT as(a) Using the one-dimensional MATLAB function fft to implement the above result, and for the signalwith values M =
The convolution sum is a fast way to find the coefficients of the polynomial resulting from the multiplication of two polynomials.(a) Suppose x[n] = u[n] - u[n - 3] find its Z-transform X(z), a
A filter has a transfer function(a) Find the poles and zeros of H(z1; z2).(b) For what values of ẑ1 and ẑ2 is H(ẑ1, ẑ2) = 0/0?(c) Ignoring the numerator, i.e., lettingis this filter BIBO
Consider a system represented by the convolution sum(a) Obtain the BIBO stability condition for this system, assuming that the input x[m, n] is bounded.(b) Use the variable transformation
Use the binomial theoremto express the transfer functionasand determine the impulse response of the system. (z + y)* = E ()-"2 y"= =ng mi-n2 n2=0 H(21, 2) = 1/(1 – (2ī' + 2,')
Consider the separable two-dimensional Z-transformwhere p0 and p1 are poles of X(z1, z2).(a) Determine the poles and zeros of X(z1, z2).(b) Carefully indicate the ROC and the location of the unit
Detection of edges is a very important application in image processing. Taking the gradient of a two-dimensional function detects the changes the edges of an image. A filter than is commonly used in
Use the function imread, rgb2gray and double to read the color image peppers.png. Convert it into a gray level image I[m, n] with double precision. Add noise to it using the function randn (Gaussian
Let the inputand the impulse response of an FIR filter be(a) Use the two-dimensional convolution function conv2 to find the output y[m; n].(b) Is the impulse response separable, i.e., h[m; n] =
A discrete-time system has a unit impulse response h[n].(a) Let the input to the discrete-time system be a pulse x[n] = u[n] - [n - 4] compute the output of the system in terms of the impulse
Echoing in music — An effect similar to multi-path in acoustics is echoing or reverberation. To see the effects of an echo in an acoustic signal consider the simulation of echoes on the handel.mat
The impulse responsesatisfies the difference equationwith zero boundary conditions. By definition(a) Use the difference equation to show that(b) Show that for m ≥ 0; n > 0determine the values of
Consider the line impulseswhere -∞ < m < ∞ and -∞ < n < ∞.(a) Draw the line impulses x[m; n] and y[m; n]. Determine if they are separable.(b) Consider the product z[m; n] = x[m;
For the two-dimensional signals(a) Draw their supports, and express these domains in terms of u1[m; n].(b) Let z[m, n] = x[m, n] - y[m, n], and draw its support. r[m, n] = am+"u12[m, n] m+ru14[m, n]
The input of an LTI continuous-time system is x(t) = u(t) - u(t – 3.5). The system’s impulse response is h(t) = u(t) - u(t – 2.5).(a) Find the system’s output y(t) by graphically computing
Suppose we would like to send the two messages mi(t), i = 1; 2, created in Problem 12 using the same bandwidth and to recover them separately. To implement this consider the QAMapproach where the
The signal at the input of an AM receiver is u(t) = m1(t) cos(20t) + m2(t) cos(100t) where the messages mi(t), i = 1; 2 are the outputs of a low–pass Butterworth filter with inputsx1(t) = r(t) -
Consider the transmission of a sinusoid x(t) = cos(2f0t) through a channel affected by multi-path and Doppler. Let there be two paths, and assume the sinusoid is being sent from a moving object so
Control systems attempt to follow the reference signal at the input, but in many cases they cannot follow particular types of inputs. Let the system we are trying to control have a transfer function
Suppose you would like to obtain a feedback implementation of an all pass system with transfer function(a) Determine the feed–forward transfer function G(s) and the feedback transfer function H(s)
Consider a filter with frequency responseor a sinc function in frequency.(a) Find the impulse response h(t) of this filter. Plot it and indicate whether this filter is a causal system or not.(b)
The Fourier transforms of even and odd functions are very important. Let x(t) = e-|t| andy(t) = e-tu(t) - etu(-t).(a) Plot x(t) and y(t), and determine whether they are even or odd.(b) Show that the
An inverted sawtooth signal is given by the reflection x(-t), where x(t) is the sawtooth signal. Use the entry for the sawtooth signal in Table 4.2 to obtain a zeromean inverted sawtooth signal y(t)
Consider the entries in Table 4.2 for the sawtooth and the triangular signals normalized in magnitude and period. Use their Fourier coefficients to obtain corresponding periodic signals that are
The periodic impulse signal with period T1 ishas Fourier coefficients Xk = 1 = T1. Suppose y1(t) is a pulse of support 0 ≤ t ≤ T1, determine the convolution y(t) = x(t) * y1(t) using the above
We wish to obtain a discrete approximation to a sinusoid x(t) = sin(3πt) from 0 to 2.5 seconds. To do so a discretized signal x(nTs), with Ts = 0:001, is multiplied it by a causal window w(nTs) of
The discretized approximation of a pulse is given bywhere N = 10000 and Ts = 0:001 seconds.(a) Obtain this signal and let the plotted signal using plot be the analog signal. Determine the duration of
Consider the following continuous-time signalCarefully plot x(t) and then find and plot the following signals:(a) x(t + 1), x(t - 1) and x(-t)(b) 0:5[x(t) + x(-t)] and 0:5[x(t) - x(-t)](c) x(2t) and
Algebra of complex numbers - Consider complex numbers z = 1 + j, w = -1 + j, v = -1 - j and u = 1 - j. You may use MATLAB compass to plot vectors corresponding to complex numbers to verify your
Exponentials — The exponential x(t) = eαt for t ≥ 0 and zero otherwise is a very common continuoustime signal. Likewise, y(n) = αn for integers n ≥ 0 and zero otherwise is a very common
Suppose you wish to find the area under a signal x(t) using sums. You will need the following result found above(a) Consider first x(t) = t, 0 ≤ t ≤ 1, and zero otherwise. The area under this
Three laws in the computation of sums arefor any permutation p(k) of the set of integers k in the summation.(a) Explain why the above rules make sense when computing sums. To do that considerLet c be
Another definition for the finite difference is the backward difference:(Δ1[x(nTs)]=Ts approximates the derivative of x(t)).(a) Indicate how this new definition connects with the finite difference
Let Find y(t) analytically and determine a value of Ts for which(consider as possible values Ts = 0:01 and Ts = 0:1). Use the MATLAB function diff or create your own to compute the finite
To get an idea of the number of bits generated and processed by a digital system consider the following applications:(a) A compact disc (CD) is capable of storing 75 minutes of “CD quality”
A phasor can be thought of as a vector, representing a complex number, rotating around the polar plane at a certain frequency in radians/second. The projection of such a vector onto the real axis
Consider a function of z = 1 + j1, w = ez(a) Find (i) log(w), (ii) Re(w), (iii) Im(w)(b) What is w + w*, where w* is the complex conjugate of w?(c) Determine |w|, ∠w and | log(w)|2 ?(d)
Consider the calculation of roots of an equation zN = α where N ≥ 1 is an integer and α = |α|ejϕ a nonzero complex numb(a) First verify that there are exactly N roots for this equation and that
Design a causal low-pass FIR digital filter with N = 21. The desired magnitude response of the filter isand the phase is zero for all frequencies. The sampling frequency fs = 2000 Hz.(a) Use a
A chirp signal is a sinusoid of continuously changing frequency. Chirps are frequently used to jam communication trans-missions. Consider the chirp(a) A measure of the frequency of the chirp is the
To design a three-band discrete spectrum analyzer for audio signals, we need to design a low-pass, a band-pass, and a high-pass IIR filters. Let the sampling frequency be Fs =10 kHz. Consider the
Consider the following continuous-time signal 1-t 0st
Given the causal full-wave rectified signal x(t) = ∣sin(2πt)∣u(t)(a) Find the even component of x(t), call it xe(t) and plot it. Is xe(t) periodic? if so, what is its fundamental period Te?
The following problems relate to the periodicity of signals:(a) Determine the frequency Ω0 in rad/sec, the corresponding frequency f0 in Hz, and the fundamental period T0 sec of these signals
In the following problems find the fundamental period of signals and determine periodicity.(a) Find the fundamental period of the following signals, and verify it(i) x(t) = cos(t + π/4),
The following problems are about energy and power of signals.(a) Plot the signal x(t) = et u(t) and determine its energy. What is the power of x(t)?(b) How does the energy of
Consider the periodic signal x(t) = cos(πt) of fundamental period T0 =2 sec.(a) Is the expanded signal x(t/2) periodic? if periodic indicate its fundamental period.(b) Is the compressed signal
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) u(t 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) u(t 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)/kT1)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 Ωhp = 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) = eat[u(t) u(t 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 Ï0? 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] = δ[n] + 2δ[n 1] + 3δ[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) = z2(zejÏ/2)(zejÏ/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(z1), 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 −
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