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computer science
signals and systems

Questions and Answers of Signals and Systems

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 jÎ© 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 jÎ© 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) ˆ’ 2u(t ˆ’ 1) + u(t ˆ’ 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) ˆ’ 2r(t ˆ’ 1) + r(t ˆ’ 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 ˆ— 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) ˆ’ u(t ˆ’ 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) eˆ’at 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) = eˆ’Î±tu(t) ˆ’
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) ˆ’ u(t ˆ’ 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(Ï€t)is a €œclipped€ 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) = ˆ’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) ˆ’ w(t ˆ’ 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(2Ï€t) 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) = Î´(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
In wireless communications, the effects of multi-path significantly affect the quality of the received signal. Due to the presence of buildings, cars, etc. between the transmitter and the receiver
Consider the sampling signalwhich we will use in the sampling of analog signals later on.(a) Plot Î´T(t). Findand carefully plot it for all t. What does the resulting signal ss(t)
When defining the impulse or Î´(t) signal the shape of the signal used to do so is not important. Whether we use the rectangular pulse we considered in this Chapter or another pulse, or
(a) Consider the periodic signals x1 (t) = 4cos(πt) and x2(t) = −sin(3πt +π/2). Find the periods T1 of x1 (t) and T2 of x2 (t) and determine if x(t) = x1 (t) + x2(t) is periodic. If so,
Signal energy and RC circuit€”The signal x(t) = eˆ’ˆ£tˆ£is defined for all values of t.(a) Plot the signal x(t) and determine if this signal is finite
Consider the triangular train of pulses x(t) in Figure 1.23.(a) Carefully plot the derivative of x(t), y(t) = dx(t)/dt.(b) Can you computeIf so, what is it equal to? If not, explain why not.(c)
For a complex exponential signal x(t) =2ej2πt(a) Suppose y(t) = ejπt, would the sum of these signals z(t) = x(t) + y(t) be also periodic? If so, what is the fundamental period of z(t)?(b) Suppose
A periodic signal can be generated by repeating a period.(a) Find the function g(t), defined in 0 ‰¤ t ‰¤ 2 only, in terms of basic signals and such that when repeated
Consider the signal x(t) in Figure 1.21.(a) Plot the even-odd decomposition of x(t), i.e., find and plot the even xe(t) and the odd xo(t) components of x(t).(b) Show that the energy of a
Let x(t) = t[u(t) − u(t −1)], we would like to consider its expanded and compressed versions.(a) Plot x(2t) and determine if it is a compressed or expanded version of x(t).(b) Plot
A signal x(t) is defined as x(t) = r(t + 1) ˆ’ r(t) ˆ’2u(t) + u(t ˆ’ 1).(a) Plot x(t) and indicate where it has discontinuities. Compute y(t) = dx(t)/dt and plot
The signalcan be written as x(t) = ˆ£tˆ£p(t).(a) Carefully plot x(t) and define p(t), then find y(t) = dx(t)/dt and carefully plot it.(b) Calculateand comment on how your
Is it true that (if not true, give correct answer)(a) for any positive integer k(b) for a periodic signal x(t) of fundamental period T0for any value of t0? Consider, for instance, x(t)
Consider the periodic signal x(t) = cos(2Ω0t) + 2 cos(Ω0t),−∞ < t < ∞, and Ω0 = π. The frequencies of the two sinusoids are said to be harmonically related.(a) Determine the
Consider a circuit consisting of a sinusoidal source vs(t) = cos(t) u(t) connected in series to a resistor Rand an inductor L and assume they have been connected for a very long time.(a) Let R =
Do reflection and time-shifting commute? That is, do the two block diagrams in Figure 1.20 provide identical signals, i.e., is y(t) equal to z(t)? To provide an answer to this consider the signal
The following problems relate to the symmetry of the signal:(a) Consider a causal exponential x(t) = eˆ’t u(t). i. Plot x(t) and explain why it is called
Consider a finite support signal x(t) = t, 0 ≤ t ≤1, and zero elsewhere.(a) Plot x(t +1) and x(−t +1). Add these signals to get a new signal y(t). Do it graphically and verify your results

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