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computer science
signals and systems
Questions and Answers of
Signals and Systems
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(kΩ0t)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=jΩ or Ω = (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(Ω) =
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 Ω1 = Ω0, 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) 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 Ω0= Ï, has a periodThe signal x(t) is the input of an ideal low-pass filter with the frequency response H(jΩ) 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 Ω0 = 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),x(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) cos(6Ï t),x2(t) = 1 + cos(2Ï t)
The input-output equation for an analog averager islet x(t) = ejΩ0t. Since the system is LTI then the output should be y(t) = ejΩ0t H(jΩ0).(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 = 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 δ(t)and dδ(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
The input/output equation for an analog averager is given by the convolution integralwhere x(t)is the input and y(t)the output.(a) Change the above equation to determine the impulse response
The steady-state solution of stable systems is due to simple poles in the jΩ axis of the s-plane coming from the input. Suppose the transfer function of the system is(a) Find the
The transfer function of a BIBO stable and causal system has poles only on the open left-hand s-plane (excluding the jΩ axis).(a) Let the transfer function of a system beand let
In convolution problems the impulse response h(t)of the system and the input x(t)are given and one is interested in finding the output of the system y(t). The so-called
Consider the following problems related to the convolution integral.(a) The impulse response of a LTI system is h(t) = e−2tu(t) and the system input is a pulse x(t) = u(t) − u(t − 3). Find
Consider the following cases where we want to determine different types of responses.(a) The input to a LTI system is x(t) = u(t) 2u(t 1) + u(t 2) and the
Consider the cascade of two LTI systems, see Figure 3.21, where the input of the cascade is z(t) and the output is y(t), while x(t)is the output of the first system and the input of the second
A wireless channel is represented by y(t) = αx(t − T) + α3x(t − 3T) where 0 < α < 1 is the attenuation and T the delay. The input is x(t) and the output y(t).(a) Find the impulse
The impulse response of an LTI is h(t) = r(t) 2r(t 1) + r(t 2) and the input is a sequence of impulses(a) Find the system output y(t)as the convolution
A causal LTI system has a transfer function(a) Find the poles and zeros of H(s), and from this determine if the filter is BIBO stable or not.(b) Draw a block diagram for such a system.(c)
The transfer function of a LTI system is(a) Use the Laplace transform to find the unit-step response s(t) = (h x) (t).(b) Find the response due to the following inputs(i) x1(t) =
You are given the following Laplace transform of the output y(t)of a system with input x(t)and Laplace transform X(s):(a) If x(t) = u(t), find the zero-state response yzs(t).(b) Find the
A system is represented by the following ordinary differential equationwhere y(t)is the system output and x(t)is the input.(a) Find the transfer function H(s) = Y(s)/X(s)of the system. From its
In the following problems we use the inverse Laplace transform and the relation between input and output of LTI systems.(a) The Laplace transform of the output of a system isfind y1(t), assume
To find the Laplace transform of x(t)=r(t)−2r(t−1)+2r(t−3)−r(t−4).(a) Plot x(t). Calculate dx(t)/dt, d2x(t)/dt2 and plot them.(b) Use the Laplace transform of d2x(t)/dt2 to obtain
Find the Laplace transform of the following signals and their region of convergence:(a) the reflection of the unit-step signal, i.e., u(−t). And then use the result together with the Laplace
Consider the pulse x(t) = u(t) u(t 1). Find the zeros and poles of X(s)and plot them.(a) Suppose x(t)is the input of a LTI system with a transfer function H(s) =
In the following problems properties of the Laplace transform are used.(a) Show that the Laplace transform of x(t) eat u(t)is X(s + a), where X(s) = L[x(t)] and then use it to find the
Find the Laplace transform of the following signals and in each case determine the corresponding region of convergence:(a) the signal x(t) = eαtu(t)
Consider the following cases involving sinusoids:(a) Find the Laplace transform of y(t) = sin(2π t)[u(t) − u(t − 1)]and its region of convergence. Carefully plot y(t). Determine the region
Find the Laplace transform of the following(a) anti-causal signals indicating their region of convergence:(i) x(t) = etu(−t),
Find the Laplace transform of the following(a) finite support signals, and indicate their region of convergence:(i) x(t) = δ(t − 1), (ii) y(t) = δ(t + 1) − δ(t − 1)(iii) z(t) = u(t
The impulse response of an ideal low-pass filter is h(t) = sin(t)/t.(a) Given that the impulse response is the response of the system to an input x(t) = δ(t) with zero initial conditions, can
The impulse response of a LTI is h(t) = e−2tu(t). Use MATLAB functions to approximate the convolution integral when the inputs of the system arex1(t) = cos(2π t) [u(t) − u(t − 20)], x2(t) =
An analog averager is given by(a) Let x(t) = u(t) u(t 1) find the average signal y(t) using the above integral. Let T = 1. Carefully plot y(t). Verify your result by
A zener diode is such that the output corresponding to an input vs(t) = cos(Ït)is a clipped sinusoidas shown in Figure 2.21 for a few periods. Use MATLAB to
The following op-amp circuit is used to measure the changes of temperature in a system (Figure 2.20). The output voltage is given byvo(t) = R(t)vi(t)Suppose that the temperature in the
The bounded-input bounded-output stability assumes that the input is always bounded, limited in amplitude. If that is not the case, even a stable system would provide an unbounded output. Consider
An echo system could be modeled using or not using feedback.(a) Feedback systems are of great interest in control and in modeling of many systems. An echo is created as the sum of one or more
The impulse response h(t) of a causal, linear, time-invariant continuous-time system isAssuming zero initial conditions, determine the outputs yi(t), i = 1,2, of this system if the input is(a) x1(t)
A quadrature amplitude modulation (QAM) system is a communication system capable of transmitting two messages m1(t), m2(t) at the same time. The transmitted signal s(t) iss(t) = m1(t) cos(Ωct) +
Consider a system represented by a first-order ordinary differential equation:(a) Show first that for a function f(t)using the definition of the derivative.(b) Apply the above result to show
The input x(t) and the corresponding output y(t) of a linear time-invariant (LTI) system arex(t) = u(t) − u(t − 1) → y(t) = r(t) − 2r(t − 1) + r(t − 2)Determine the outputs yi(t), i = 1,
The input-output equation characterizing a system of input x(t) and output y(t) isand zero otherwise.(a) Find the ordinary differential equation that also characterizes this system.(b) Suppose x(t) =
The response of a first-order system is for t ¥ 0and zero otherwise.(a) Consider y(0) = 0 is the system linear? If y(0) 0, is the system linear? Explain.(b) If x(t) = 0, what
Consider the system where for an input x(t) the output is y(t) = x(t)f(t).(a) Let f(t) = u(t) − u(t − 10), determine whether the system with input x(t) and output y(t) is linear, time-invariant,
Consider the analog averager,where x(t) is the input and y(t) the output.(a) Find the impulse response h(t) of the averager. Is this system causal?(a) Let x(t) = u(t), find the output of the
An analog system has the following input-output relation,and zero otherwise. The input is x(t) and y(t) is the output.(a) Is this system LTI? If so, can you determine without any computation the
A time-varying capacitor is characterized by the charge-voltage equation q(t) = C(t) v(t). That is, the capacitance is not a constant but a function of time.(a) Given that i(t) = dq(t)/dt, find
An RC circuit in series with a voltage source x(t) is represented by a ordinary differential equationwhere y(t) is the voltage across the capacitor. Assume y(0) is the initial voltage across the
(a) A system is represented by the ordinary differential equation dz(t)/dt = w(t) w(t 1) where w(t) is the input and z(t) the output.i. How is this system related to an
The input-output relationship of a system iswhere x(t) is the input and y(t) the output.(a) Let the input be x(t) = sin(2Ït) u(t), plot the corresponding output y(t). What is the
Consider an averager represented by the input/output equationwhere x(t) is the input and y(t) the output.(a) Let the input be x1(t) = δ(t), find graphically the corresponding output
Consider the full-wave rectified signaly(t) = |sin(πt)| −∞(a) As a periodic signal, y(t) does not have finite energy but it has a finite-power Py. Find it.(b) It is always useful to get a quick
One of the advantages of defining the δ(t) functions is that we are now able to find the derivative of discontinuous signals. Consider a periodic sinusoid defined for all timesx(t) = cos(Ω0 t)
An interesting phenomenon in the generation of musical sounds is beating or pulsation. Suppose NP different players try to play a pure tone, a sinusoid of frequency 160 Hz, and that the signal
Consider now the Doppler effect in wireless communications. The difference in velocity between the transmitter and the receiver causes a shift in frequency in the signal, which is called the Doppler
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