Questions and Answers of Signals and Systems

Given that the Z-transform of a discrete-time cosine A cos(Ï‰0n) 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) ˆ’ 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) = eˆ’tu(t) and a disturbance input v(t) = (1ˆ’eˆ’t) 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) ˆ’ 2u(t) + u(t ˆ’ 1) and let(a) Plot x(t) and y(t)(b) Find X(Î©) and carefully plot its magnitude spectrum. Is X(Î©)
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(Î©)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(Î©) 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) ˆ’ 2u(t ˆ’ 1) + u(t ˆ’ 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 Î´Ts(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(Î©) (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(jÎ©) of this system by
The Fourier series coefficients of a periodic signal x(t), with fundamental frequency Î©0= Ï€/4 are X1= Xˆ—ˆ’1=j, X5= Xˆ—ˆ’5= 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(jÎ©)|of the filter. Indicate the magnitude
The Fourier transform of a signal x(t) is(a) Carefully plot X(Î©)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) ˆ’ 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(Î©) = F(s(t)).(b) Find the magnitude and phase of S(Î©).(c) Use the equivalent expression
A periodic signal x(t) has a periodx1(t) = r(t) ˆ’ 2r(t ˆ’ 1) + r(t ˆ’ 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)
In the following problems we want to find the Fourier transform of the signals.(a) For the signalfind its Fourier transform by using the Fourier transform ofx(t) = 0.5e€“at u(t) ˆ’ 0.5eat
Use properties of the Fourier transform in the following problems.(a) Use the Fourier transform of cos(kÎ©0t)to find the Fourier transform of a periodic signal x(t) with a Fourier
The derivative property can be used to simplify the computation of some Fourier transforms. Letx(t) = r(t) − 2r(t − 1) + r(t − 2)(a) Find and plot the second derivative with respect to tof
Find the Fourier transform of δ(t − τ) and use it to find the Fourier transforms of(a) δ(t − 1) + δ(t + 1).(b) cos(Ω0t).(c) sin(Ω0t).
Starting with the Fourier transform pairand using no integration, indicate the properties of the Fourier transform that will allow you to compute the Fourier transform of the following signals (do
The Fourier transform of a signal x(t) isUse properties of the Fourier transform to(a) find the integral(b) Find the value x(0).(c) Let s=jÎ© or Î© = (s/j), find X(s) and
There are signals whose Fourier transforms cannot be found directly by either the integral definition or the Laplace transform. For instance, the sinc signalis one of them.(a) Let X(Î©) =
A causal signal x(t) having a Laplace transform with poles in the open-left s-plane (i.e., not including the jΩ-axis) has a Fourier transform that can be found from its Laplace transform. Consider
As seen in the previous problem, the Fourier series is one of a possible class of representations in terms of orthonormal functions. Consider the case of the Walsh functions which are a set of
We wish to approximate the triangular signal x(t), with a period x1(t) = r(t) − 2r(t − 1) + r(t − 2), by a Fourier series with a finite number of terms, let’s say 2N. This approximation
Consider a full-wave rectifier that has as output a periodic signal x(t) of fundamental period T0= 1 and a period of it is given as(a) Obtain the Fourier coefficients Xk.(b) Suppose we pass x(t)
The smoothness of a period deter-mines the way the magnitude line spectrum decays. Consider the following periodic signals x(t) and y(t) both of fundamental period T0 = 2sec., and with a period from
Rectifying a sinusoid provides a way to create a dc source. In this problem we consider the Fourier series of the full and the half-wave rectified signals. The full-wave rectified signal xf(t) has a
The computation of the Fourier series coefficients is helped by the relation between the formula for these coefficients and the Laplace transform of a period of the periodic signal.(a) A periodic
Let x(t) = sin2(2πt), a periodic signal of fundamental period T0 = 0.5, and y(t) = |sin(2π t)|also periodic of the same fundamental period.(a) A trigonometric identity gives that x(t) = 0.5[1 −
We want to use the Fourier series of a train of square pulses (done in the chapter) to compute the Fourier series of the triangular signal x(t)with a periodx1(t) = r(t) − 2r(t − 1) + r(t −
Consider the Fourier series of two periodic signals(a) Let Î©1 = Î©0, is z(t) = x(t) y(t) periodic? If so, what is its fundamental period and its Fourier series
Consider the integral of the Fourier series of the pulse signal p(t) = x(t) ˆ’1 of period T0= 1, where x(t) is given in Figure 4.22.(a) Given that an integral of p(t) is the area under the
Given the Fourier series representation for a periodic signal x(t), we can compute derivatives of it just like with any other signal.(a) Consider the periodic train of pulses shown in Figure
The following problems are about the steady state response of LTI systems due to periodic signals.(a) The transfer function of a LTI system isIf the input to this system is x(t) = 1 + cos(t +
Consider the periodic signal x(t) shown in Figure 4.21.(a) Use the Laplace transform to compute the Fourier series coefficients Xk, k‰ 0 of x(t).(b) Suppose that to find the
A periodic signal x(t), of fundamental frequency Î©0= Ï€, has a periodThe signal x(t) is the input of an ideal low-pass filter with the frequency response H(jÎ©) shown
We are interested in designing a dc voltage source. To do so, we full-wave rectify an AC voltage to obtain the signal x(t) =|cos(πt)|,−∞(a) Specify the magnitude response |H(jΩ)|of the
Consider the following problems related to filtering of periodic signals:(a) A periodic signal x(t)of fundamental frequency Î©0 = Ï€/4 is the input of an ideal band-pass filter
Consider a periodic signal x(t) with fundamental period T0 =1 and a period x1(t) = −0.5t[u(t) − u(t − 1)].(a) Consider the derivative g(t) = dx(t)/dt. Indicate if g(t) is periodic and if
Consider the following problems related to steady state and frequency responses.(a) The input x(t)and the output y(t),ˆ’ˆžx(t) = 4 cos(2Ï€ t) + 8 sin(3Ï€
A period of a periodic signal x(t)with fundamental period T0 = 2 isx1 (t) = cos(t) [u(t) − u(t − 2)].(a) Plot the signal x(t) and find the Fourier series coefficients {Xk} for x(t) using the
Consider a periodic signalx(t) = 0.5 + 4 cos(2π t) − 8 cos(4π t) − ∞(a) Determine the fundamental frequency Ω0 of x(t).(b) Find the Fourier series
Let the Fourier series coefficients of a periodic signal x(t) of fundamental frequency Ω0 =2π/T0 be {Xk}. Consider the following functions of x(t).y(t) = 2x(t)−3, z(t) = x(t − 2) +
Suppose you have the Fourier series of two periodic signals x(t)and y(t)of fundamental periods T1 and T2, respectively. Let Xk and Yk be the Fourier series coefficients corresponding to x(t) and
Consider the following problems related to the exponential Fourier series.(a) The exponential Fourier series of a periodic signal x(t)of fundamental period T0 isi. Determine the value of the
Let a periodic signal x(t)with a fundamental frequency Ω0 = 2π have a periodx1(t) = t[u (t) − u(t − 1)](a) Plot x(t), and indicate its fundamental period T0.(b) Compute the Fourier
Find the complex exponential Fourier series for the following signals. In each case plot the magnitude and phase line spectra for k ≥ 0. (i)x1(i) x1(t) = cos(5t + 45°), (ii) x2(t) =
A periodic signal x(t) has a fundamental frequency Ω0 = 1 and a period of it isx1(t) = u(t) − 2u(t − π) + u(t − 2π)(a) Find the Fourier series coefficients {Xk} of x(t) using their
Consider the following problems related to periodicity and Fourier series:(a) For the signalsx1(t) = 1 + cos(2Ï€ t) ˆ’ cos(6Ï€ t),x2(t) = 1 + cos(2Ï€ t)
The input-output equation for an analog averager islet x(t) = ejÎ©0t. Since the system is LTI then the output should be y(t) = ejÎ©0t H(jÎ©0).(a) Find the integral
The eigenfunction property is only valid for LTI systems. Consider the cases of non-linear and of time-varying systems.(a) A system represented by the input-output equation y(t) = x2(t) is
Consider a negative feedback system used to control a plant with transfer functionG(s) = 1/(s(s + 1)(s + 2)).The output y(t)of the feedback system is connected via a sensor with transfer function
When the numerator or denominator are given in a factorized form, we need to multiply polynomials. Although this can be done by hand, MATLAB provides the function convthat computes the coefficients
In the generation of dc from ac voltage, the ”half-wave” rectified signal is an important part. Suppose the ac voltage is x(t) = sin(2π t) u(t) and y(t) is the half-wave rectified
Consider the convolution of a pulse x(t) = u(t + 0.5) − u(t − 0.5) with itself many times. Use MATLAB for the calculations and the plotting.(a) Consider the result for N = 2 of these
Consider the following functions Yi(s) = L[yi(t)], i = 1, 2 and 3,where {yi(t), i = 1, 2, 3} are the complete responses of differential equations with zero initial conditions.(a) For each of
An analog averager can be represented by the differential equationwhere y(t)is its output and x(t)the input.(a) If the input-output equation of the averager isShow how to obtain the above
The type of transient you get in a second-order system depends on the location of the poles of the system. The transfer function of the second-order system isand let the input be x(t) =
Let Y(s) = L[y(t)] be the Laplace transform of the solution of a second-order differential equation representing a system with input x(t) and some initial conditions,(a) Find the zero-state
The following function Y(s) = L[y(t)] is obtained applying the Laplace transform to a differential equation representing a system with non-zero initial conditions and input x(t), with Laplace
One of the uses of the Laplace trans-form is the solution of differential equations.(a) Suppose you are given the ordinary differential equation that rep-resents a LTI system,y(2)(t) + 0.5y(1)
The poles corresponding to the Laplace transform X(s)of a signal x(t) are p1,2 = ˆ’3 ±jÏ€/2 and p3=0.(a) Within some constants, give a general form of the signal
Consider the following inverse Laplace problems:(a) Use MATLAB to compute the inverse Laplace transform ofDetermine the value of x(t)in the steady state. How would you be able to obtain this
Consider the following inverse Laplace transform problems.(a) Given the Laplace transformwhich is not proper, determine the amplitude of the Î´(t)and dÎ´(t)/dt terms in
The following problems consider approaches to stabilize an unstable system.(a) An unstable system can be stabilized by using negative feedback with a gain Kin the feedback loop. For instance
There are two types of feedback, negative and positive. In this problem we explore their difference.(a) Consider negative feedback. Suppose you have a system with transfer function H(s) =
To see the effect of the zeros on the complete response of a system, sup-pose you have a system with a transfer function(a) Find and plot the poles and zeros of H(s). Is this BIBO

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