Problem1. The input x[n] and the output y[n] of a system are related by the equation

y[n]=x[n-1]+x[1-n].

Is the system time invariant (yes/no)? Justify your answer.

Solution. No:

x[n]→ [time delay n0] →y[n]=x[n-n0] → [system] →z[n]=y[n-1]+y[1-n] =x[n-n0-1]+x[1-n-n0]=x[(n-1)-n0]+x[(1-n)-n0]

x[n]→[system] →y[n]=x[n-1]+x[1-n] →[time delay n0] →z[n]=y[n-n0]= x[n-n0-1]+x[1-(n-n0)]=x[(n-1)-n0]+x[(1-n)+n0]

Problem2. (from book 1.27) In this chapter, we introduced a number of general properties of systems.

In practular, a system may or may not be

a)memoryless

b)Time invariant

3)linear

4)causal

5)stable

Determine which of these properties hold and which do not hold for each of the following continuous-

time systems. Justify your answers. In each example, y(t) denotes the system output and x(t) is the

system input.

a)y(t)=x(t-2)+x(2-t) b)y(t)=cos(3t)x(t)

c)

Solution.

This is not causal as y(t) depends on x(t+h) in calculating dx(t)/dt. If we put x(t)=sqrt(t) then

y(t)=1/(2sqrt(t)) which is unbounded for t=0. Then the system is not stable.

Problem3. (problem from text book 1.28) Determine which of the properties listed in problem 1.27

hold and which do not hold for each of the following discrete-time systems. Justify your answers. In

each example, y[n] denotes the system output and x[n] is the system input.

Solution.

Problem4. (from text book 1.3) Determine the values of the power and energy for each the following

systems:

Solution.

Problem5. (problem 2.11 from text-book)

Let x(t)=u(t-3)-u(t-5) and h(t)=e-3tu(t)

a)Compute y(t)=x(t)*h(t)

b)Compute g(t)=(dx(t)/dt)*h(t)

c)How is g(t) is related to y(t)?

Solution.

Problem6. (problem 2.21 from text book) Compute the convolution y[n]=x[n]*h[n] of the following pairs

of signals:

a)x[n]=αnu[n]

h[n]= βnu[n]

here α≠β

b) x[n]=h[n]=αnu[n]

c) x[n]=(-1/2)nu[n-4]

h[n]= 4nu[2-n]

d) x[n] and h[n] are as in Figure : x[n]

0 1 2 3 4

h[n]

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Amplitude of each signal is 1.

Solution.

Problem7.

Problem8.

Problem9.

Solution.

Solution.

Problem 10.( problem 2.30 from text book)

Consider the first-order difference equation.

y[n]+2y[n-1]=x[n]

Assuming the condition of initial rest(i.e. şf x[n]=0 for n<n0, then y[n]=0 for n<n0), find the impulse

response of a system whose input and output are related by this difference equation. You may solve the

problem by rearranging the difference equation so as to express y[n] in terms of y[n-1] and x[n] and

generating the values of y[0], y[1], y[2],… in that order.

Solution.

Problem11. (Problem 2.29 (a,d,f) from text book )

The following are the impulse of CT LTI system. Determine whether each system is causal and/or stable.

a)h(t)=e-4tu(t-2)

d)h(t)=e2tu(-1-t)

f)h(t)=te-tu(t)

Solution.

a)

Problem12.

Find the exponential Fourier series for a signal x(t ) = cos(5t )sin (3t ) . You can do this

without evaluating any integrals.

Solution.