VE216 Homework1 Solved

Problems:

1.    [6!] Consider the periodic sinusoidal signal illustrated below.

x(t)

 

(a)    Find the mathematical representation for this signal.

(b)    Find the energy of this signal. Is it an energy signal, power signal, or neither?

(c)    Carefully sketch and find a mathematical expression for the output signal of an integrator system,

i.e., where x(t) is under the interval [−π,π].

2.    [12!] Determine the values of average power and energy for each of the following signals:

 

                                                                                                                                       {e−t           t 0,

3.    [3!] Find the average value, power, and energy of signal x(t) =

                                                                                                                                         0         otherwise.

4.    [6!] Consider the signal illustrated below.

 

(a)    Find a mathematical representation for x(t).

(b)    Sketch s(t) = x(−2t+1)/2 by performing graphical time transformations. Sketch the intermediate signal each time you make a transformation, like time-shifting, or time-scaling.

(c)    Decompose x(t) into its even and odd components. Carefully sketch the even and odd components of x(t).

5.       [6!] A discrete-time signal x[n] is shown below. Sketch and label carefully each of the following signals:

(a)    x[3n]

(b)    x[n]u[3 − n]

 

6.       [9!] Suppose x1(t) and x2(t) are periodic signals with fundamental periods T1 0 and T2 0 respectively.

(a)    Show that if T1/T2 is rational, then x(t) = x1(t) + x2(t) is periodic.

(b)    Similarly, show that if T1/T2 is rational, then x(t) = x1(t)x2(t) is periodic and the least common multiple of T1 and T2 is a period of x(t).

(c)    Determine whether the following signals are periodic. If so, find a period. Otherwise, specify the reason.

 

7.       [4!] Considering the signals

 

Show that z(t) = x(t)y(t) is periodic, and write z(t) as a linear combination of harmonically related complex exponentials. That is, find a number T and complex numbers Ck such that

z(t) = ∑ckejk(2π/T)t

k

8.       [3!] Some warm ups for MATLAB. Use MATLAB to plot the following three signals.

(a)    y(t) = et

(b)    y(t) = e−0.1tsin(πt)

(c)    y(t) = sin(πt + π/4)

9.       [4!] Consider the signal illustrated below.

 

(a)    Express the signal x(t) using a sum of step functions.

(b)    Find the derivative of the signal and carefully sketch it.

10.   [10!] Indicate whether the following systems are Memoryless, Time Invariant, Linear, Causal, Stable. Justify your answers. (3! for each)

 

11.   [8!] A linear system H has following input-output pairs. Answer the following question, and justify your answers.

                    x1(t)                                                                                                                                  y1(t)

                                                                                       .            

 

                       0          1                       t                                   H                                                             0          1                                    t

                    x2(t)                                                                                                                                  y2(t)

                        0          1          2          3           t                       H                                                t

                    x3(t)                                                                                                                                  y3(t)

 

                       0          1          2           t                                   H                                                             0          1          2          3          t

                                                                                                                                            .

(a)    Could this system be causal?

(b)    Could this system be time invariant?

(c)    Could this system be memoryless?

(d)   What is the output for the input x4(t), sketch it.

12.   [3!] Prove that the product of two odd signals is an even signal.

13.     [6!] Let s(t) = (t − 1)2rect(t − 1)

                                             2                     2

(a)    Make a sketch of s(t).

(b)    Evaluate , where 

14.   [2!] Show that causality for a continuous-time linear system is equivalent to the following statement:

For any time t0 and any input x(t) such that x(t) = 0 for t < t0, the corresponding output y(t) must also be zero for t < t0.

15.   [6!] A system has the input and output relation given by

y[n] = Tx[n] = nx[n].

Is the system

(a)    linear?

(b)    time invariant?

(c)    bounded input bounded output (BIBO) stable?

(d)   memoryless?

(e)    causal?

16.   [8!] Let x[n] be a discrete-time signal, and let

y1[n] = x[2n]

and

                                                                                                    {x[n/2],       n even

y2[n] =

                                                                                                       0,                 n odd

The signals y1[n] and y2[n] respectively represent in some sense the speeded up and slowed down versions of x[n]. However, it should be noted that the discrete-time notions of speeded up and slowed down have subtle differences with respect to their continuous-time counterparts. Consider the following statements:

(a)    If x[n] is periodic, then y1[n] is periodic.

(b)    If y1[n] is periodic, then x[n] is periodic.

(c)    If x[n] is periodic, then y2[n] is periodic. (d) If y2[n] is periodic, then x[n] is periodic.

For each of these statements, determine whether it is true, and if so, determine the relationship between the fundamental periods of the two signals considered in the statement. If the statement is not true, produce a counterexample to it.

17.   [4!] Given a signal x(t),

(a)    suppose it is an energy signal with energy E[x(t)] = Ex. Then what is the energy of the signal x(−at + b), i.e. E[x(−at + b)]?

(b)    suppose it is a power signal with power P[x(t)] = Px. Then what is the power of the signal x(−at + b), i.e. P[x(−at + b)]?