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computer science
signals and systems
Questions and Answers of
Signals and Systems
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] x1[n 2] + x1[n 4] where x1[n] = u[n] u[n 2]. The
An LTI system represented by the difference equation y[n] = y[n 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 = 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[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) u(t 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 z1terms in the numerator, z1can 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]= αn(u[n] u[n 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] x[n 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] u[n 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) + η(t)where s(t) is the desired signal and η(t) is additive noise. From experience, we know that the average power of the desired
The transfer function of a discrete-time system iswith α = r1 ejθ1 and β = r2ejθ2, where ri > 0 and θi are angles between 0 and
Suppose we cascade a differentiator and a smoother 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 . 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 ≥
Given that the Z-transform of a discrete-time cosine A cos(Ï0n) u[n] is(a) Use the given Z-transform to find a difference equation whose output y[n] is a discrete-time cosine A
Two systems with transfer functionsare connected in parallel.(a) Use MATLAB to determine the transfer function H(z)of the over-all system.(b) Use the function tf2ssto obtain state-variable
We are interested in the unit-step solu-tion of a system represented by the following difference equation y[n] = y[n − 1] − 0.5y[n − 2] + x[n] + x[n − 1](a) Find an expression for
The Pade approximant provides an exact matching of M + N – 1 values of h[n], where M and N are the orders of the numerator and denominator of the rational approximation. But there is no
Consider finding the inverse Z-transform of(a) MATLAB does the partial fraction expansion as:while we do it in the following form:Show that the two partial fraction expansions give the
Suppose that a state realization has the following matricesFind the corresponding transfer function and verify it using the function ss2tf. Obtain a minimal realization of this system. Draw a block
Suppose you are given the observer space representation with matrix and vectorsTo find a transformation that diagonalizes Ao use MATLAB function eigswhich calculates the eigenvalues and eigenvectors
For the system in Part 1, consider the state variablesv1(t) = y(t) v2(t)=ẏ(t) + y(t) − x(t)(a) Obtain the matrix A2 and the vectors b2 and cT2 for the state and the output equations that
A LTI system is represented by an ordinary differential equation(a) Obtain the transfer function H(s) = Y(s)/X(s) = B(s)/A(s) and find its poles and zeros. Is this system BIBO stable? Is there
Given the two realizations in Figure 6.27 obtain the corresponding transfer functionsFigure 6.27: Y1 (s) Н () %— X1 (s) Y,(s) -, and H2(s) X,(s) Realization 1 y1 (t) v1(t) v2(t) r1(t) Realization
You are given a state-variable realization of a second-order system hav-ing the following matrix and vectors(a) Find an invertible matrix F that can be used to transform the given state and
Let the transfer function of a system beShow that by defining the state-variables asv1(t) = y(t),v2(t)=Ë y(t) + a1y(t) b1x(t)we obtain a minimal state variable and output
To explore the performance of a proportional-plus-derivative controller on a second-order system, let Gp(s) = 1/(s(s + 1)) be the transfer function of the plant and Gc(s) = K1 + K2s be the
Consider a second-order system with transfer functionwhere Y(s) and X(s) are the Laplace transforms of output y(t) and the input x(t) of the system. Q is called the quality factor.(a) If the
The feedback system shown in Figure 6.26 has two inputs: the conventional input x(t) = etu(t) and a disturbance input v(t) = (1et) u(t).(a) Find the transfer
Consider the cascade connection of two continuous-time systems shown in Figure 6.25where(a) Determine the input/output differential equation for the overall cascade connection.(b) Suppose that w(0) =
Consider the cascade of two continuous-time systems shown in Figure 6.24. The input-output characterization of system A is x(t) = dz(t)/dt. It is known that system B is linear and time-invariant, and
Consider the following problems connected with the feedback system shown in Figure 6.23.(a) The transfer function of the plant in Figure 6.23 is G(s) = 1/(s(s + 1)). If we want the impulse
Let H(s) = Y(s)/X(s) be the transfer function of the feedback system in Figure 6.22. The impulse response of the plant (with transfer function Hp(s)) is hp(t) = sin(t) u(t).(a) If we want the
The feed forward transfer function of a negative feedback system is G(s) = N(s)/D(s), and the feedback transfer function is unity. Let X(s) be the Laplace transform of the input x(t) of the feedback
A resistor R = 1Ω, a capacitor C = 1 F and an inductor L = 1 H are connected in series with a source vi(t). Consider the output the voltage across the capacitor vo(t).(a) Use integrators and
Consider a series RC circuit with input a voltage source vi(t) and output the voltage across the capacitor vo(t).(a) Draw a negative feedback system for the circuit using an integrator, a
The transfer function H(s) = 1/(s + 1)2of a filter is to be implemented by cascading two first order filters Hi(s) = 1/(s + 1), i = 1, 2.(a) Implement Hi(s) as a series RC circuit with input vi(t)
Consider the following filters with the given poles and zeros, and dc constantH1(s): K = 1 poles p1 = −1, p2,3 = −1 ± jπzeros z1 = 1,z2,3 = 1 ± jπH2(s): K = 1 poles p1 = −1, p2,3 = −1 ±
Consider an RLC series circuit with a voltage source vs(t). Let the values of the resistor, capacitor, and inductor be unity. Plot the poles and zeros and the corresponding frequency responses of the
Consider the signal x(t) = u(t + 1) 2u(t) + u(t 1) and let(a) Plot x(t) and y(t)(b) Find X(Ω) and carefully plot its magnitude spectrum. Is X(Ω)
The smoothness of the signal determines the frequency content of its spectrum. Consider the signalsx(t) = u(t + 0.5) − u(t − 0.5),y(t) = (1 + cos(π t))[u(t + 0.5) − u(t − 0.5)](a) Plot these
The supports in time and in frequency of a signal x(t) and its Fourier transform X(Ω)are inversely proportional. Consider a pulse(a) Let T0 = 1, and T0 = 10 and find and compare the
The connection between the Fourier series and the Fourier transform can be seen by considering what happens when the fundamental period of a periodic signal increases to a point at which the
A pure tone x(t) = 4 cos(1000t) is transmitted using an amplitude mod-ulation communication system with a carrier cos(10000t). The output of the AM system isy(t) = x(t) cos(10000t)At the receiver,
Suppose you want to design a dc-source using a half-wave rectified signal x(t) and an ideal filter. Let x(t) be periodic, T0= 2, and with a period(a) Find the Fourier transform X(Ω) of
An analog averager is characterized by the following relationshipwhere x(t) is the input and y(t) the output. If x(t) = u(t) 2u(t 1) + u(t 2),(a) find the
The sampling signalwill be important in the sampling theory later on.(a) As a periodic signal of fundamental period Ts express δTs(t) by its Fourier series.(b) Determine then
As indicated by the derivative property, if we multiply a Fourier transform by (jΩ)N it corresponds to computing an Nth derivative of its time signal. Consider the dual of this property. That is, if
If the Fourier transform of the pulse x(t) given in Figure 5.14 is X(Ω) (do not need to compute it)(a) Using the properties of the Fourier transform (no integration needed) obtain the
A continuous-time LTI system is represented by the ordinary differential equationwhere x(t) is the input and y(t) the output.(a) Determine the frequency response H(jΩ) of this system by
The Fourier series coefficients of a periodic signal x(t), with fundamental frequency Ω0= Ï/4 are X1= X1=j, X5= X5= 2 and the
Consider the cascade of two filters with frequency responsesH1(jΩ) = jΩ, and H2(jΩ) = 1e−jΩ(a) Indicate what each of the filters does.(b) Suppose that the input to the cascade isx(t) = p(t)
The transfer function of a filter is(a) Find the poles and zeros of H(s) and use this information to sketch the magnitude response |H(jΩ)|of the filter. Indicate the magnitude
The Fourier transform of a signal x(t) is(a) Carefully plot X(Ω)as function of .(b) Determine the value of x(0). [u(2 +T) – u(S2 – 1)] X(2)
The frequency response of an ideal low-pass filter is(a) Calculate the impulse response h(t) of the ideal low-pass filter.(b) If the input of the filter is a periodic signal x(t) having a
Consider the raised cosine pulsex(t) = [1 + cos(π t)] (u(t + 1) − u(t − 1))(a) Carefully plot x(t).(b) Find the Fourier transform of the pulsep(t) = u(t + 1) − u(t − 1)(c) Use the
The following problems relate to the modulation property of the Fourier transform:(a) Consider the signalx(t) = p(t) + p(t) cos(Ï t) where p(t) = u(t + 1) u(t
A sinc signal x(t) = sin(0.5t)/(πt)is passed through an ideal low-pass filter with a frequency response H(jΩ) = u(Ω + 0.5) − u(Ω − 0.5)(a) Find the Fourier transform X(Ω)and
Consider the sign signal(a) Find the derivative of s(t) and use it to find S(Ω) = F(s(t)).(b) Find the magnitude and phase of S(Ω).(c) Use the equivalent expression
A periodic signal x(t) has a periodx1(t) = r(t) 2r(t 1) + r(t 2), T0 = 2(a) Find the Fourier series of z(t) = d2x(t)/dt2 using the Laplace
Consider the following problems related to the modulation and power properties of the Fourier transform.(a) The carrier of an AM system is cos(10t), consider the following message signalsi. m(t)
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