VE216 Homework2 Solved

Problems:

1.    [3!] Assume a system has the input-output relationship y(t) = f(t)x(t), where x(t) is the input and y(t) is the output. f(t) is not constant, i.e., there exists t0,t1 that f(t0) ≠ f(t1). Show that this system is timevariant. That is, a system with time-variant gain cannot be time-invariant. (Hint: find a counterexample.)

2.    [9!]Here are input-output relationships for a few systems, all of which are linear. Some of them are timeinvariant, some are not. Determine which are which. Find the impulse response of the time-invariant systems.

 ,

 ,

 .

Hint: Note that if you can transform the above relationships into the exact form of convolution y(t) = g(t) ∗ x(t), then the system is immediately time-invariant with g(t) being the impulse response h(t). That is because different from algebraic operators like multiplication, the convolution operator implies time-invariance itself.

3.    [9!]One of following two statements is correct, and the other is incorrect. The symbol ∗ denotes convolution.

•    If y(t) = h(t) ∗ x(t) then y(t − 3) = h(t − 3) ∗ x(t − 3);

•    Or if y(t) = h(t) ∗ x(t) then y(t − 3) = h(t) ∗ x(t − 3).

(a)    Give a simple proof of the correct statement.

(b)    Give a simple counterexample for the incorrect statement.

(c)    Repeat (a) and (b) for the following two statements. The symbol · denotes multiplication.

•    If y(t) = h(t) · x(t) then y(t − 3) = h(t − 3) · x(t − 3);

•    Or if y(t) = h(t) · x(t) then y(t − 3) = h(t) · x(t − 3).

Be careful with the notation h(t) ∗ x(t). More precise notation is (h ∗ x)(t), which makes it clear that convolution is an operation on two signals, not a point-wise operation like multiplication.

4.    [8!] Often we will be convolving two signals that are zero everywhere except over some finite range (called finite support signals). Suppose x1(t) is non-zero over the range a ≤ t ≤ b and that x2(t) is non-zero over the range c ≤ t ≤ d. Suppose y(t) = x1(t) ∗ x2(t).

(a)    Find the range of values of t for which y(t) is possibly non-zero.

(b)    Compute rect((t − 2)/2) ∗ rect((t + 3)/4) (express answer with braces and carefully sketch). Check your result with part (a).

5.       [6!] For each of the following pairs of waveforms, use convolution integral to find the response y(t) of the LTI system with impulse response h(t) and input x(t). Sketch your results.

(a)    x(t) = e−αtu(t),h(t) = e−βtu(t) (Do this both when α = β and α ≠          β)

(b)    x(t) and h(t) as in the figure below, the slope for the straight line is a and the line intersects y-axis at (0,b):

 

6.       [6!]

(a)    Consider a linear system with input x(t) and output y(t) given by

 .

Is this system time-invariant?

(b)    Consider another LTI system. Let its impulse response h(t) be the triangular pulse shown below, and x(t) be the impulse train

 ,

SKETCH y(t) = x(t) ∗ h(t) for T = 4,2,1.5 and 1.(No formulae are needed though you still want to label your graphs clearly.)

h(t)

 

7.       [6!] Let y(t) = (x ∗ h)(t). Show the following properties of convolution.

 ,

 ,

8.       [6!] Compute the following convolution:

(a)    u(t) ∗ u(t),

(b)    u(t) ∗ t2u(t).

9.       [3!] Suppose you have a ”black box” LTI system and you discover that for a unit-step input signal the response of the system is the function (3 − t)rect(t−21). Determine the impulse response of the system. (Hint: See Problem 7.)

10.   [6!] Determine whether the following systems are linear, stable, causal, time-invariant, and memoryless.

 

11.   [6!] Consider an LTI system whose response to the signal x1(t) is the signal y1(t) which are illustrated below.

(a)    Determine and sketch carefully the response of the system to the input x2(t) depicted below. (b) Determine and sketch carefully the response of the system to the input x3(t) depicted below.

 

 

12.   [6!] The triangular pulse is defined as tri(t) = (1 − |t|)rect(t/2).

Compute x(t) = tri(t/2) ∗ rect(t−21).Express your answer using braces, and carefully sketch.

13.   [6!] Find the impulse response of the following LTI systems and further determine whether they are causal, stable and static.

 

14.   [3!] Consider an LTI system S and a signal x(t) = 2e−3tu(t − 1). If

x(t) → y(t)

and

determine the impulse response  

15.   [12!] We are given a certain LTI system with impulse response h0(t). We are told that when the input is x0(t) the output is y0(t), which is sketched below. We are then given the following set of inputs to LTI systems with the indicated impulse responses.

                                                                         Input x(t)                                Impulse response h(t)

                                                             (a)       x(t) = 2x0(t)                            h(t) = h0(t)

 (b)

(c)                                 0 −   0

(d)                                x(t) = x0(−t)  h(t) = h0(−t)

(e)                                 x(t) = x′0(t)   h(t) = h0(t)

(f)                                  x(t) = x′0(t)   h(t) = h′0(t)

In each of these cases, determine whether or not we have enough information to determine the output y(t) when the input is x(t) and the system has impulse response h(t). If so, provide an accurate sketch of it with numerical values clearly indicated on the graph.

 

16.   [5!] Find the expression of response of the CT system described by the linear constant-coefficient differential equation.