EE 3054: Signals, Systems and Transforms Problems

EE 3054: Signals, Systems and Transforms

Problems

Ivan W. Selesnick

December 18, 2004

Contents

1 Discrete-Time Signals and Systems 21.1 Basic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 More Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Z Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Frequency Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6 Inverse Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.7 Summary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.8 Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.9 More Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2 Continuous-Time Signals and Systems 402.1 Basic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.3 LaplaceTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.5 Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.6 FourierTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.7 Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.8 FourierSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.9 Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3 The Sampling Theorem 80

1 Discrete-Time Signals and Systems

1.1 Basic Problems

1.1.1 Make an accurate sketch of each of the signals

x(n) = u(n + 3) + 0.5 u(n− 1)

x(n) = δ(n + 3) + 0.5 δ(n− 1)

x(n) = 2n · δ(n− 4)

x(n) = 2n · u(−n− 2)

x(n) = (−1)n u(−n− 4).

x(n) = 2 δ(n + 4)− δ(n− 2) + u(n− 3)

x(n) =∞∑

4 δ(n− 3 k − 1)

x(n) =∞∑

k=−∞

(−1)k δ(n− 3 k)

1.1.2 Sketch x(n), x1(n), x2(n), and x3(n) where

x(n) = u(n + 4)− u(n), x1(n) = x(n− 3),

x2(n) = x(5− n), x3(n) =n∑

k=−∞

1.1.3 Sketch x(n) and x1(n) where

x(n) = (0.5)n u(n), x1(n) =n∑

k=−∞

1.1.4 Sketch x(n) and x1(n) where

x(n) = n [δ(n− 5) + δ(n− 3)], x1(n) =n∑

k=−∞

1.1.5 A discrete-time system may be classified as follows:

• memoryless/with memory

• causal/noncausal

• linear/nonlinear

• time-invariant/time-varying

• BIBO stable/unstable

Classify each of the following discrete-times systems.

y(n) = cos(x(n)).

y(n) = 2 n2 x(n) + n x(n + 1).

y(n) = max {x(n), x(n + 1)}

y(n) ={

x(n) when n is evenx(n− 1) when n is odd

y(n) = x(n) + 2 x(n− 1)− 3 x(n− 2).

y(n) =∞∑

(1/2)k x(n− k).

That is,

y(n) = x(n) + (1/2) x(n− 1) + (1/4) x(n− 2) + · · ·

y(n) = x(2 n)

1.1.6 The impulse response of a discrete-time LTI system is given by

h(n) ={

1 if n is a positive prime number0 otherwise

(a) Is the system causal?

(b) Is the system BIBO stable?

1.1.7 You observe an unknown system (not necessarily LTI) and notice that

u(n)− u(n− 5) - S - (0.4)n u(n)

and that

u(n)− u(n− 10) - S - (0.2)n u(n)

Which conclusion can you make?

(a) The system is LTI.

(b) The system is not LTI.

(c) There is not enough information to decide.

1.1.8 You observe an unknown system (not necessarily LTI) and notice that

n2 u(n) - S - (0.5)n u(n)

To each of the following questions, answer yes, no, or it can not be determined.

(a) Is the system stable?

(b) Is the system causal?

(c) Is the system time-invariant?

1.1.9 You observe an unknown LTI system and notice that

u(n)− u(n− 2) - S - δ(n− 1)− 14 δ(n− 4)

Sketch the step response s(n). The step response is the system output when the input is the stepfunction u(n).

1.1.10 The impulse response of a discrete-time LTI system is

h(n) = 2 δ(n) + 3 δ(n− 1) + δ(n− 2).

Find and sketch the output of this system when the input is the signal

x(n) = δ(n) + 3 δ(n− 1) + 2 δ(n− 2).

1.1.11 Consider a discrete-time LTI system described by the rule

y(n) = x(n− 5) +12

x(n− 7).

What is the impulse response h(n) of this system?

h(n) = δ(n) + 2 δ(n− 1) + δ(n− 2).

Sketch the output of this system when the input is

x(n) =∞∑

δ(n− 4 k).

h(n) = 2 δ(n)− δ(n− 4).

Find and sketch the output of this system when the input is the step function

x(n) = u(n).

1.1.14 Consider the discrete-time LTI system with impulse response

h(n) = n u(n).

(a) Find and sketch the output y(n) when the input x(n) is

x(n) = δ(n)− 2 δ(n− 5) + δ(n− 10).

(b) Classify the system as BIBO stable/unstable.

1.1.15 The impulse response h(n) of an LTI system is given by

h(n) =(

Find and sketch the output y(n) when the input is given by

(a) x(n) = δ(n)

(b) x(n) = δ(n− 2)

1.1.16 Consider the LTI system with impulse response

h(n) = δ(n)− 14

δ(n− 2).

(a) Find and sketch the output y(n) when the input x(n) is

x(n) = 2−n u(n).

1.1.17 For an LTI system it is known that input signal

x(n) = δ(n) + 3 δ(n− 1)

produces the following output signal:

y(n) =(

What is the output signal when the following input signal is applied to the system?

x2(n) = 2 δ(n− 2) + 6 δ(n− 3)

h(n) = δ(n− 1).

(a) Find and sketch the output y(n) when the input x(n) is the impulse train with period 6,

x(n) =∞∑

k=−∞

δ(n− 6k).

1.1.19 An LTI system is described by the following equation

y(n) =∞∑

x(n− k).

Sketch the impulse response h(n) of this system.

1.1.20 Consider the parallel combination of two LTI systems.

- h2(n)

- h1(n)

l+ - y(n)

You are told that

h1(n) = u(n)− 2 u(n− 1) + u(n− 2).

You observe that the step response of the total system is

s(n) = 2 r(n)− 3 r(n− 1) + r(n− 2)

where r(n) = n u(n). Find and sketch h2(n).

1.2 More Convolution

1.2.1 Derive and sketch the convolution x(n) = f(n) ∗ g(n) where

f(n) = 2 δ(n + 10) + 2 δ(n− 10)

g(n) = 3 δ(n + 5) + 3 δ(n− 5)

f(n) = δ(n− 4)− δ(n− 1)

g(n) = 2 δ(n− 4)− δ(n− 1)

f(n) = −δ(n + 2)− δ(n + 1)− δ(n)

g(n) = δ(n) + δ(n + 1) + δ(n + 2)

f(n) = 4

g(n) = δ(n) + 2 δ(n− 1) + δ(n− 2).

f(n) = δ(n) + δ(n− 1) + 2 δ(n− 2)

g(n) = δ(n− 2)− δ(n− 3).

f(n) = (−1)n

g(n) = δ(n) + δ(n− 1).

h(n) = u(n)− u(n− 5).

Sketch the output of this system when the input is

x(n) =∞∑

δ(n− 5 k).

1.2.3 The N -point moving average filter has the impulse response

h(n) ={

1/N 0 ≤ n ≤ N − 10 otherwise

Use the Matlab conv command to compute

y(n) = h(n) ∗ h(n)

for N = 5, 10, 20, and in each case make a stem plot of h(n) and y(n).

What is the general expression for y(n)?

1.2.4 The convolution of two finite length signals can be written as a matrix vector product. Look atthe documentation for the Matlab convmtx command and the following Matlab code that shows theconvolution of two signals by (1) a matrix vector product and (2) the conv command. Describe theform of the convolution matrix and why it works.

>> x = [1 4 2 5]; h = [1 3 -1 2];

>> convmtx(h’,4)*x’

1713921-110

>> conv(h,x)’

1713921-110

1.3 Z Transforms

1.3.1 The Z-transform of the discrete-time signal x(n) is

X(z) = −3 z2 + 2 z−3

Find and sketch the signal x(n).

1.3.2 Define three discrete-time signals:

a(n) = u(n)− u(n− 4)b(n) = δ(n) + 2 δ(n− 3)c(n) = δ(n)− δ(n− 1)

Define three new Z-transforms:

D(z) = A(−z), E(z) = A(1/z), F (z) = A(−1/z)

(a) Sketch a(n), b(n), c(n)

(b) Write the Z-transforms A(z), B(z), C(z)

(c) Write the Z-transforms D(z), E(z), F (z)

(d) Sketch d(n), e(n), f(n)

1.3.3 Define the discrete-time signal x(n) as

x(n) = −0.3 δ(n + 2) + 2.0 δ(n) + 1.5 δ(n− 3)− δ(n− 5)

(a) Sketch x(n).

(b) Write the Z-transform X(z).

(c) Define G(z) = z−2 X(z). Sketch g(n).

1.3.4 Let x(n) be the length-5 signal

x(n) = {1, 2, 3, 2, 1}

where x(0) is underlined. Sketch the signal corresponding to each of the following Z-transforms.

(a) X(2z)

(b) X(z2)

(c) X(z) + X(−z)

(d) X(1/z)

1.3.5 Determine the discrete-time signal x(n) with the Z-transform

X(z) = (1 + 2 z) (1 + 3 z−1) (1− z−1).

h(n) = 3(

Find the output y(n) when the input x(n) is

x(n) =(

1.3.7 The transfer function of a discrete-time LTI system is

H(z) = 2 + 4 z−1 − z−3.

(a) Sketch the impulse response h(n).(b) Find the difference equation to describe this system.(c) Classify the system as BIBO stable/unstable.

1.3.8 Sketch the step response of the system in the previous problem.(In other words, sketch the output signal when the input signal is the step function u(n).)

What is the steady state value of the step response (as n →∞)?

1.3.9 Consider the transfer functions of two discrete-time LTI systems,

H1(z) = 1 + 2z−1 + z−2,

H2(z) = 1 + z−1 + z−2.

(a) If these two systems are cascaded in series, what is the impulse response of the total system?

x(n) - H1(z) - H2(z) - y(n)

(b) If these two systems are combined in parallel, what is the impulse response of the total system?

- H2(z)

- H1(z)

j+ - y(n)

1.3.10 Consider the parallel combination of two LTI systems.

- h2(n)

- h1(n)

l+ - y(n)

You are told that the impulse responses of the two systems are

h1(n) = 3(

h2(n) = 2(

(a) Find the impulse response h(n) of the total system.

(b) You want to implement the total system as a cascade of two first order systems g1(n) and g2(n).Find g1(n) and g2(n), each with a single pole, such that when they are connected in cascade, theygive the same system as h1(n) and h2(n) connected in parallel.

x(n) - g1(n) - g2(n) - y(n)

1.3.11 Consider the cascade combination of two LTI systems.

x(t) - SYS 1 - SYS 2 - y(t)

The impulse response of SYS 1 is

h1(n) = δ(n) + 0.5 δ(n− 1)− 0.5 δ(n− 2)

and the transfer function of SYS 2 is

H2(z) = z−1 + 2 z−2 + 2 z−3.

(a) What is the impulse response of the total system?

(b) What is the transfer function of the total system?

1.3.12 Consider a stable discrete-time LTI system described by the difference equation

y(n) + 2 y(n− 1) = x(n)− x(n− 1).

(a) Find the transfer function H(z) and its ROC.

(b) Find the impulse response h(n).

1.4 Difference Equations

1.4.1 A causal discrete-time system is described by the difference equation,

y(n) = x(n) + 3 x(n− 1) + 2 x(n− 4)

(a) What is the transfer function of the system?

(b) What is the impulse response of the system?

1.4.2 Given the impulse response h(n) of a causal discrete-time LTI system,

h(n) = δ(n)− 12

δ(n− 2)

(a) For the input x(n) = 3n, what is the output y(n)?

(b) For the input x(n) = an, what is the output y(n)?

(c) What is the transfer function H(z)?

(d) What is the difference equation describing this system?

1.4.3 Given two discrete-time LTI systems described by the difference equations

T1 : y(n) +13

y(n− 1) = x(n) + 2x(n− 1)

T2 : y(n) +13

y(n− 1) = x(n)− 2x(n− 1)

let T be the cascade of T1 and T2 in series. What is the difference equation describing T? Suppose T1

and T2 are causal systems. Is T causal? Is T stable?

1.4.4 Consider the parallel combination of two causal discrete-time LTI systems.

- SYS 1

- SYS 2

l+ - r(n)

You are told that System 1 is described by the difference equation

y(n)− 0.1 y(n− 1) = x(n) + x(n− 2)

and that System 2 is described by the difference equation

y(n)− 0.1 y(n− 1) = x(n) + x(n− 1).

Find the difference equation of the total system.

1.4.5 Consider a causal discrete-time LTI system described by the difference equation

y(n)− 56

y(n− 1) +16

y(n− 2) = 2 x(n) +23

x(n− 1).

(a) Find the transfer function H(z).

(b) Find the impulse response h(n). You may use Matlab to do the partial fraction expansion. Seethe command residue. Make a stem plot of h(n) with Matlab.

(c) Plot the magnitude of the frequency response |H(ejω)| of the system. Use the Matlab commandfreqz.

1.4.6 Consider a causal discrete-time LTI system with the impulse response

h(n) =32

u(n) + 2 δ(n− 4)

(a) Make a stem plot of h(n) with Matlab.

(b) Find the transfer function H(z).

(c) Find the difference equation that describes this system.

(d) Plot the magnitude of the frequency response |H(ejω)| of the system. Use the Matlab commandfreqz.

1.4.7 Consider a stable discrete-time LTI system described by the difference equation

y(n) + 2 y(n− 1) = x(n)− x(n− 1).

(a) Find the transfer function H(z) and its ROC.

(b) Find the impulse response h(n).

1.4.8 A room where echos are present can be modeled as an LTI system that has the following rule:

y(n) =∞∑

2−k x(n− 10 k)

The output y(n) is made up of delayed versions of the input x(n) of decaying amplitude.

(a) Sketch the impulse response h(n).

(b) What is transfer function H(z)?

(c) Write the corresponding finite-order difference equation.

1.5 Frequency Responses

1.5.1 The frequency response Hf (ω) of a discrete-time LTI system is

Hf (ω) ={

e−jω −0.4π < ω < 0.4π0 0.4π < |ω| < π.

x(n) = 1.2 cos(0.3 π n) + 1.5 cos(0.5 π n).

Put y(n) in simplest real form (your answer should not contain j).

Hint: Use Euler’s formula and the relation

ejωon −→ LTI SYSTEM −→ Hf (ωo) ejωon

1.5.2 The frequency response Hf (ω) of a discrete-time LTI system is as shown.

−π 0

Hf (ω)

��H

x(n) = 1 + cos(0.3 π n).

Put y(n) in simplest real form (your answer should not contain j).

1.5.3 A stable linear time invariant system has the transfer function

H(z) =z + 2

(z − 1/2)(z + 4)

Find the frequency response of this system.

Find the output of this system for the input x(n) = cos(0.2π n).

1.5.4 A causal LTI system is implemented with the difference equation

y(n) = 3 x(n)− 2 x(n− 1)− 0.5 y(n− 1) + 0.1 y(n− 2).

Find the frequency response of this system. Using Matlab, make a plot of |Hf (ω)|.Find the output of this system for the input x(n) = cos(0.2π n).

1.5.5 The mangitude and phase of the frequency response of a discrete-time LTI system are:

|Hf (ω)| ={

2 for |ω| < 0.5 π1 for 0.5π < |ω| < π.

∠Hf (ω) ={

0.3 π for − π < ω < 0−0.3 π for π < ω < 0.

Find the output signal y(n) when the input signal is

x(n) = 2 sin(0.2 π n) + 3 cos(0.6 π n + 0.2 π).

1.6 Inverse Systems

1.6.1 A causal discrete-time LTI system

x(n) - H(z) - y(n)

is described by the difference equation

y(n)− 13

y(n− 1) = x(n)− 2 x(n− 1).

What is the impulse response of the stable inverse of this system?

1.6.2 Consider a discrete-time LTI system with the impulse response

h(n) = δ(n + 1)− 103

δ(n) + δ(n− 1).

Find the impulse response g(n) of the stable inverse of this system.

h(n) = −δ(n) + 2(

Find the impulse response of the stable inverse of this system.

1.6.4 A causal discrete-time LTI system

x(n) - h(n) - y(n)

has the impulse response

h(n) = δ(n) + 3.5 δ(n− 1) + 1.5 δ(n− 2).

(a) Give the difference equation that describes this system.

(b) Find the impulse response of the stable inverse of this system.

1.7 Summary Problems

h(n) =(

x(n) = u(n)− u(n− 2).

Simplify your mathematical formula for y(n) as far as you can. Show your work.

h(n) = δ(n) + δ(n− 2).

x(n) =∞∑

k=−∞

(−1)k δ(n− 2 k).

Simplify your mathematical formula for y(n) as far as you can. Show your work.

h(n) = u(n)− u(n− 5).

(a) What is the transfer function H(z) of this system?

(b) What difference equation implements this system?

1.7.4 A causal LTI system is implemented using the difference equation

y(n) = x(n) +914

y(n− 1)− 114

y(n− 2)

(a) What is the transfer function H(z) of this system?

(b) What is the impulse response h(n) of this system?

1.7.5 The signal g(n) is given by

g(n) =(

u(n− 1)

(a) Sketch g(n).

(b) Find the Z-transform G(z) of the signal g(n) and its region of convergence.

h(n) ={

0.25 for 0 ≤ n ≤ 30 for other values of n.

Make an accurate sketch of the output of the system when the input signal is

x(n) ={

1 for 0 ≤ n ≤ 300 for other values of n.

1.7.7 Two discrete-time LTI systems have the following impulse responses

h1(n) =(

u(n), h2(n) =(−1

If the two systems are connected in parallel,

- H2(z)

- H1(z)

j+ - y(n)

Find and make an accurate sketch of the impulse response of the total system.

1.7.8 If a discrete-time LTI system has the transfer function H(z) = 5, then what difference equationimplements this system? Classify this system as memoryless/with memory.

1.7.9 First order difference system

A discrete-time LTI system is implemented using the difference equation

y(n) = 0.5 x(n)− 0.5 x(n− 1).

(a) What is the transfer function H(z) of the system?

(b) What is the impulse response h(n) of the system?

(c) What is the frequency response Hf (ω) of the system?

(d) Accurately sketch the frequency response magnitude |Hf (ω)|.(e) Find the output y(n) when the input signal is the step signal u(n).

(f) Sketch the pole-zero diagram of the system.

(g) Is the system a low-pass filter, high-pass filter, or neither?

1.7.10 A causal discrete-time LTI system is implemented with the difference equation

y(n) = 3 x(n) +32

y(n− 1).

(a) Find the output signal when the input signal is

x(n) = 3 (2)n u(n).

Show your work.

(b) Is the system stable or unstable?

(c) Sketch the pole-zero diagram of this system.

1.8 Matching

1.8.1 The diagrams on the following pages show the impulse responses, frequency responses, and pole-zerodiagrams of 4 causal discrete-time LTI systems. But the diagrams are out of order. Match each diagramby filling out the following table.

In the pole-zero diagrams, the zeros are shown with ‘o’ and the poles are shown by ‘x’.

IMPULSE RESPONSE FREQUENCY RESPONSE POLE-ZERO DIAGRAM

MatchA

0 10 20 30 40−1

−0.5

IMPULSE RESPONSE 1

0 10 20 30 40

−0.5

IMPULSE RESPONSE 2

0 10 20 30 400

1.2IMPULSE RESPONSE 3

0 10 20 30 40

−0.5

IMPULSE RESPONSE 4

−1 −0.5 0 0.5 10

FREQUENCY RESPONSE 2

−1 −0.5 0 0.5 10

−1 −0.5 0 0.5 1

−0.5

ZERO−POLE DIAGRAM 3

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

1.8.2 The diagrams on the following pages show the impulse responses, frequency responses, and pole-zerodiagrams of 4 causal discrete-time LTI systems. But the diagrams are out of order. Match each diagramby filling out the following table.

IMPULSE RESPONSE FREQUENCY RESPONSE POLE-ZERO DIAGRAM

MatchB

0 10 20 30 40

−0.5

IMPULSE RESPONSE 1

0 10 20 30 40

IMPULSE RESPONSE 2

0 10 20 30 400

−1 −0.5 0 0.5 10

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

1.8.3 The diagrams on the following pages show the impulse responses and pole-zero diagrams of 8 causaldiscrete-time LTI systems. But the diagrams are out of order. Match each diagram by filling out thefollowing table.

IMPULSE RESPONSE POLE-ZERO DIAGRAM

12345678

SecondOrderMatching/Match1

0 10 20 30 40

IMPULSE RESPONSE 7

0 10 20 30 40

IMPULSE RESPONSE 8

0 10 20 30 40

−0.5

IMPULSE RESPONSE 1

0 10 20 30 40

−0.5

IMPULSE RESPONSE 6

0 10 20 30 40−1

−0.5

IMPULSE RESPONSE 5

0 10 20 30 400

IMPULSE RESPONSE 2

0 10 20 30 400

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

1.8.4 The diagrams on the following pages show the frequency responses magnitudes |Hf (ω)| and pole-zerodiagrams of 8 causal discrete-time LTI systems. But the diagrams are out of order. Match each diagramby filling out the following table.

FREQUENCY RESPONSE POLE-ZERO DIAGRAM

12345678

SecondOrderMatching/Match2

−1 −0.5 0 0.5 10

−1 −0.5 0 0.5 10.5

−1 −0.5 0 0.5 10

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

1.8.5 The following figure shows the frequency response magnitudes |Hf (ω)| of four discrete-time LTI systems(Systems A, B, C, and D). The signal

x(n) = 2 cos(0.15 π n) u(n− 5) + 2 cos(0.24 π n) u(n− 5)

shown below is applied as the input to each of the four systems. The input signal x(n) and each ofthe four output signals are also shown below. But the output signals are out of order. For each of thefour systems, identify which signal is the output signal. Explain your answer.

SYSTEM OUTPUT SIGNAL

InputOutput

0 10 20 30 40 50 60 70 80 90 100−4

0 0.25 π 0.5 π 0.75 π π0

FREQUENCY RESPONSES

ω0 0.25 π 0.5 π 0.75 π π

0 0.25 π 0.5 π 0.75 π π0

ω0 0.25 π 0.5 π 0.75 π π

0 10 20 30 40 50 60 70 80 90 100−3

OUTPUT SIGNALS

0 10 20 30 40 50 60 70 80 90 100−1

−0.5

0 10 20 30 40 50 60 70 80 90 100−4

0 10 20 30 40 50 60 70 80 90 100−2

1.8.6 The diagrams on the following pages show the frequency responses, impulse responses and pole-zerodiagrams of 4 causal discrete-time LTI systems. But the diagrams are out of order. Match each diagramby filling out the following table.

FREQUENCY RESPONSE POLE-ZERO DIAGRAM IMPULSE RESPONSE

MatchC

−1 −0.5 0 0.5 10

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

0 10 20 30 40

−0.1

IMPULSE RESPONSE 2

0 10 20 30 40

−0.3

−0.2

−0.1

IMPULSE RESPONSE 4

0 10 20 30 40−0.15

−0.1

−0.05

IMPULSE RESPONSE 1

0 10 20 30 40

−0.2

IMPULSE RESPONSE 3

1.8.7 The diagrams on the following pages show the frequency responses and pole-zero diagrams of 6 causaldiscrete-time LTI systems. But the diagrams are out of order. Match each diagram by filling out thefollowing table.

FREQUENCY RESPONSE POLE-ZERO DIAGRAM

123456

MatchD

−1 −0.5 0 0.5 10

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

−1 −0.5 0 0.5 1

−0.5

1.8.8 The diagrams on the following pages show the pole-zero diagrams and impulse responses of 8 causaldiscrete-time LTI systems. But the diagrams are out of order. Match each diagram by filling out thefollowing table.

POLE-ZERO DIAGRAM IMPULSE RESPONSE

12345678

PoleZero

−1.5 −1 −0.5 0 0.5 1 1.5

−0.5

POLE−ZERO DIAGRAM 2

−1.5 −1 −0.5 0 0.5 1 1.5

−0.5

−1.5 −1 −0.5 0 0.5 1 1.5

−0.5

−1.5 −1 −0.5 0 0.5 1 1.5

−0.5

−1.5 −1 −0.5 0 0.5 1 1.5

−0.5

−1.5 −1 −0.5 0 0.5 1 1.5

−0.5

−1.5 −1 −0.5 0 0.5 1 1.5

−0.5

−1.5 −1 −0.5 0 0.5 1 1.5

−0.5

0 10 20 30 40 500

IMPULSE RESPONSE 1

0 10 20 30 40 50−2

2IMPULSE RESPONSE 5

0 10 20 30 40 50−20

20IMPULSE RESPONSE 4

0 10 20 30 40 50−2

2IMPULSE RESPONSE 8

0 10 20 30 40 50−5

5IMPULSE RESPONSE 7

0 10 20 30 40 50−1

−0.5

IMPULSE RESPONSE 3

0 10 20 30 40 50−1

−0.5

IMPULSE RESPONSE 2

0 10 20 30 40 50−1

−0.5

IMPULSE RESPONSE 6

1.9 More Problems

1.9.1 Two-Point Moving Average A discrete-time LTI system has impulse response

h(n) = 0.5 δ(n) + 0.5 δ(n− 1).

(a) Sketch the impulse response h(n).

(d) Find the frequency resposnse Hf (ω). Find simple expressions for |Hf (ω)| and ∠Hf (ω) and sketchthem.

(e) Is this a lowpass, highpass, or bandpass filter?

(f) Find the output signal y(n) when the input signal is x(n) = u(n).Also, x(n) = cos(ωo n)u(n) for what value of ωo?

(g) Find the output signal y(n) when the input signal is x(n) = (−1)n u(n).Also, x(n) = cos(ωo n)u(n) for what value of ωo?

1.9.2 Four-Point Moving Average A discrete-time LTI system has impulse response

h(n) = 0.25 δ(n) + 0.25 δ(n− 1) + 0.25 δ(n− 2) + 0.25 δ(n− 3)

(a) Sketch h(n).

(d) Find the frequency resposnse Hf (ω). Find simple expressions for |Hf (ω)| and ∠Hf (ω) and sketchthem.

(e) Is this a lowpass, highpass, or bandpass filter?

h(n) = A (0.7)n u(n).

Suppose the signal

x(n) = B (0.9)n u(n)

is input to the system. A and B are unknown constants. Which of the following could be the outputsignal y(n)? Choose all that apply and provide an explanation for your answer.

(a) y(n) = K1 (1.6)n u(n) + K2 (0.2)n u(n)

(b) y(n) = K1 (0.7)n u(−n) + K2 (0.2)n u(−n)

(c) y(n) = K1 (0.7)n u(n) + K2 (0.9)n u(n)

(d) y(n) = K1 (0.7)n u(n) + K2 (0.2)n u(−n)

(e) y(n) = K1 (0.7)n u(n) + K2 (0.2)n u(n) + K3 (0.9)n u(n)

(f) y(n) = K1 (0.7)n u(n) + K2 (0.2)n u(n) + K3 u(n)

h(n) = A (0.7)n u(n).

Suppose the signal

x(n) = B cos(0.2 π n) u(n)

is input to the system. A and B are unknown constants. Which of the following could be the outputsignal y(n)? Choose all that apply and provide an explanation for your answer.

(a) y(n) = K1 (0.7)n cos(0.2 π n + θ) u(n)

(b) y(n) = K1 (0.14)n cos(0.5 π n + θ) u(n)

(c) y(n) = K1 (0.14)n u(n) + K2 cos(0.14 π n + θ) u(n)

(d) y(n) = K1 (0.14)n u(−n) + K2 cos(0.14 π n + θ)u(n)

(e) y(n) = K1 (0.7)n u(−n) + K2 cos(0.2 π n + θ) u(n)

(f) y(n) = K1 (0.7)n u(n) + K2 cos(0.2 π n + θ) u(n)

1.9.5 A causal LTI discrete-time system is implemented using the difference equation

y(n) = b0 x(n) + b1 x(n− 1)− a1 y(n− 1)− a2 y(n− 2)

where ak, bk are unknown constants. Which of the following could be the impulse response? Chooseall that apply and provide an explanation for your answer.

(a) h(n) = K1 an u(n) + K2 bn u(n)

(b) h(n) = K1 an u(n) + K2 rn cos(ω1 n + θ) u(n)

(c) h(n) = K1 rn cos(ω1 n + θ) u(n)

(d) h(n) = K1 rn1 cos(ω1 n + θ1) u(n) + K2 rn

2 cos(ω2 n + θ2) u(n) (with ω1 6= ω2).

1.9.6 Feedback Loop Consider the following connection of two discrete-time LTI systems.

h1(n)6

You are told that

h1(n) = δ(n) + δ(n− 1), h2(n) =12

δ(n− 1).

(a) Find the impulse response h(n) of the total system.

(b) Sketch h1(n), h2(n), and h(n).

1.9.7 Feedback Loop Consider the following interconnection of two causal discrete-time LTI systems

SYS 26

where SYS 1 is implemented using the difference equation

y(n) = 2 x(n)− 0.5 y(n− 1)

and SYS 2 is implemented using the difference equation

y(n) = 2 x(n)− 0.2 y(n− 1)

Find the impulse response of the total system, htot(n).

2 Continuous-Time Signals and Systems

2.1 Basic Problems

2.1.1 Make an accurate sketch of each continuous-time signal.

x(t) = u(t + 1)− u(t),d

dtx(t),

−∞x(τ) dτ

x(t) = e−t u(t),d

dtx(t),

−∞x(τ) dτ

Hint: use the product rule for ddtx(t).

x(t) =1t

[δ(t− 1) + δ(t + 2)],∫ t

−∞x(τ) dτ

x(t) = r(t)− 2 r(t− 1) + 2 r(t− 3)− r(t− 4)

where r(t) := t u(t) is the ramp function.

g(t) = x(3− 2 t), where x(t) is defined as x(t) = 2−t u(t− 1).

2.1.2 Sketch the continuous-time signals f(t) and g(t) and the product signal f(t) · g(t).

f(t) = u(t + 4)− u(t− 4)

g(t) =∞∑

k=−∞

δ(t− 3 k)

f(t) = cos(π

g(t) =∞∑

k=−∞

δ(t− 2k)

Also write f(t) · g(t) in simple form.

f(t) = sin(π

g(t) =∞∑

k=−∞

δ(t− 2k − 1)

Also write f(t) · g(t) in simple form.

f(t) =(

g(t) =∞∑

k=−∞

δ(t− 2 k).

f(t) =∞∑

k=−∞

(−1)k δ(t− 0.5 k)

g(t) = 2−|t|

2.1.3 A continuous-time system is described by the following rule

y(t) =x(t)

x(t− 1).

Classify the system as:

(a) memoryless/with memory

(b) causal/noncausal

(c) linear/nonlinear

(d) time-invariant/time-varying

(e) BIBO stable/unstable

−1 0 1 2 3 4

��A

A- h(t) -

−1 0 1 2 3 4

��A

What is the output of the same LTI system when the input is as shown?

−1 0 1 2 3 4

��A

A��A

A- h(t) -

−1 0 1 2 3 4

2.1.5 You observe an unknown continuous-time LTI system and notice that

−1 0 1 2 3 4

��A

A- h(t) -

−1 0 1 2 3 4

��A

−1 0 1 2 3 4

��A

AAA�

- h(t) -

−1 0 1 2 3 4

Use the LTI property and be careful!

2.1.6 You observe an unknown system and notice that

u(t)− u(t− 1) - S - e−t u(t)

and that

u(t)− u(t− 2) - S - e−2t u(t)

Which conclusion can you make?

−1 0 1 2 3 4

1- S -

−1 0 1 2 3 4

��

−1 0 1 2 3 4

1- S -

−1 0 1 2 3 4

2.1.8 You observe an unknown continuous-time LTI system and notice that

u(t)− u(t− 1) - S - r(t)− 2 r(t− 1) + r(t− 2)

(a) Find and sketch the step response s(t).

(b) Find and sketch the impulse response h(t).

(c) Classify the system as BIBO stable/unstable.

2.1.9 The impulse response h(t) of an LTI system is the triangular pulse shown.

−1 0 1 2 3

��@

Suppose the input x(t) is the periodic impulse train

x(t) =∞∑

k=−∞

δ(t− k T ).

Sketch the output of the system y(t) when

(a) T = 3

(b) T = 2

(c) T = 1.5

2.1.10 Consider an LTI system described by the rule

y(t) = x(t− 5) +12

x(t− 7).

What is the impulse response h(t) of this system?

h(t) = t u(t).

(a) Find and sketch the output y(t) when the input x(t) is

x(t) = δ(t)− 2 δ(t− 2) + δ(t− 3).

h(t) = δ(t)− 14

δ(t− 2).

(a) Find and sketch the output y(t) when the input x(t) is

x(t) = 2−t u(t).

2.1.13 Consider the continuous-time LTI system with impulse response

h(t) = δ(t− 1).

(a) Find and sketch the output y(t) when the input x(t) is the impulse train with period 2,

x(t) =∞∑

k=−∞

δ(t− 2k).

2.2 Convolution

2.2.1 Derive and sketch the convolution v(t) = f(t) ∗ g(t) where f(t) and g(t) are as shown.

-−1 0 1 2 3

2.2.2 Derive and sketch the convolution x(t) = f(t) ∗ g(t) where f(t) and g(t) are as shown.

-−2 −1 0 1 2

2.2.3 Sketch the continuous-time signals f(t), g(t); find and sketch the convolution f(t) ∗ g(t).

f(t) = −u(t + 1) + u(t)g(t) = −u(t− 1) + u(t− 2).

f(t) = δ(t + 1)− δ(t− 2.5)

g(t) = 2 δ(t + 1.5)− δ(t− 2)

f(t) = δ(t) + δ(t− 1) + 2 δ(t− 2)

g(t) = δ(t− 2)− δ(t− 3).

f(t) = δ(t + 1.2)− δ(t− 1)

g(t) = δ(t + 0.3)− δ(t− 1).

f(t) = u(t + 2)− u(t− 2)

g(t) =∞∑

k=−∞

δ(t− 3 k)

f(t) = e−t u(t),

g(t) = e−t u(t)

2.2.4 The impulse response of a continuous-time LTI system is given by h(t). Find and sketch the outputy(t) when the input is given by x(t). Also, classify each system as BIBO stable/unstable.

h(t) = 2 e−3t u(t).

x(t) = u(t− 2)− u(t− 3).

h(t) ={

1− |t| |t| ≤ 10 |t| ≥ 1

x(t) =∞∑

k=−∞

δ(t− 4 k).

h(t) = e−t u(t)

x(t) = u(t)− u(t− 2),

h(t) = cos (πt)u(t).

x(t) = u(t)− u(t− 3).

2.2.5 Consider the cascade connection of two continuous-time LTI systems

x(t) - SYS 1 - SYS 2 - y(t)

with the following impulse responses,

h1(t)-

−1 0 1 2 3

h2(t)-

−1 0 3

Accurately sketch the impulse response of the total system.

2.3 LaplaceTransform

You may use MATLAB. The residue and roots commands should be useful for some of the followingproblems.

2.3.1 Given h(t), find H(s) and its region of convergence (ROC).

h(t) = 5e−4tu(t) + 2e−3tu(t)

2.3.2 Given H(s), use partial fraction expansion to expand it (by hand). You may use the Matlab commandresidue to verify your result.

H(s) =s + 4

s2 + 5s + 6

H(s) =s− 1

2s2 + 3s + 1

2.3.3 Let

f(t) = e−t u(t), g(t) = e−t u(t)

Use the Laplace transform to tind and sketch the convolution of these two signals, y(t) = f(t) ∗ g(t).(Yes, f(t) and g(t) are the same here.)

2.3.4 The impulse response of a continuous-time LTI system is given by

h(t) =(

Sketch h(t) and find the transfer function H(s) of the system.

2.3.5 Given the impulse response of a continuous-time LTI system, find the transfer function H(s), the ROCof H(s), and the poles of the system. Use Euler’s formula!

(a) h(t) = sin(3 t) u(t)

(b) h(t) = e−t/2 sin(3 t) u(t)

(c) h(t) = e−t u(t) + e−t/2 cos(3 t)u(t)

2.3.6 Suppose the impulse response of an LTI system has the form

h(t) = B e−3t u(t).

Suppose a signal x(t) with the form

x(t) = A cos(10π t) u(t)

is applied to the system. Which of the following signal forms can the output take? (Chose all thatapply.)

(a) y(t) = C e−3t cos(10π t + θ) u(t)

(b) y(t) = C cos(10π t + θ)u(t) + D e−3t u(t)

(c) y(t) = C e−3t cos(10π t + θ1) u(t) + D cos(10π t + θ2)u(t) + E e−3t u(t)

2.3.7 What is the Laplace transform of r(t− 2)?

L{r(t− 2)} = ?

where r(t) is the ramp function t u(t).

2.4 Differential Equations

2.4.1 A causal continuous-time LTI system is described by the differential equation

2 y′(t) + y(t) = 3 x(t).

(a) Find the output signal when the input signal is

x(t) = 5 e−3 t u(t).

Show your work.

(b) Is the system stable or unstable?

y′′(t) + 4 y′(t) + 3 y(t) = 2 x′(t) + 3 x(t)

(a) Find the output signal y(t) when the input signal x(t) is

x(t) = 3 e−2t u(t).

(b) What is the steady state output when the input signal is

x(t) = 3 u(t).

(The steady state output is the output signal value after the transients have died out.)

2.4.3 The impulse response h(t) of a continuous-time LTI system is given by

h(t) = 3 e−2t u(t)− e−3t u(t).

(a) Find the transfer function H(s) of the system.

(b) Find the differential equation for this system.

(c) Classify the system as stable/unstable.

2.4.4 The differential equation of a causal continuous-time LTI system is given by

y′′(t) + y′(t)− 2 y(t) = x(t)

(a) Find the transfer function H(s) of the system.

(b) Find the impulse response h(t) of this system.

h(t) = 2 e−t u(t)− e−2t u(t)

Find the differential equation that describes the system.

2.4.6 Consider a causal continuous-time LTI system described by the differential equation

y′′(t) + 3 y′(t) + 2 y(t) = 2 x′(t)

(a) Find the transfer function H(s), its ROC, and its poles.

(b) Find the impulse response h(t).

(d) Find the output of the system when the input signal is

x(t) = 2 u(t).

y′′(t) + 5 y′(t) + 4 y(t) = 3 x′(t) + 6 x(t)

(a) Find the impulse response h(t).(b) Find the output signal y(t) when the input signal x(t) is as shown:

−1 0 1 2 3

y′′(t) + y(t) = x(t).

(a) Find the transfer function H(s), its ROC, and its poles.(b) Find the impulse response h(t).(c) Classify the system as stable/unstable.(d) Find the step response of the system (the output when x(t) = u(t)).

2.4.9 A causal LTI system is described by the differential equation

y′′(t) + 7 y′(t) + 12 y(t) = 3 x′(t) + 2 x(t).

(a) Find the transfer function H(s) and the ROC of H(s).(b) List the poles of H(s).(c) Find the impulse response h(t).(d) Classify the system as stable/unstable.(e) Fnd the output y(t) when the input is

x(t) = e−t u(t) + e−2t u(t).

2.4.10 Given the two LTI systems described the the differential equations:

T1 : y′′(t) + 3 y′(t) + 7 y(t) = 2 x′(t) + x(t)T2 : y′(t)′ + y′(t) + 4 y(t) = x′(t)− 3 x(t)

(a) Let T be the cascade of T1 and T2, T [x(t)] = T2[T1[x(t)]], as shown in the diagram.

x(t) - SYS 1 - SYS 2 - y(t)

What are H1(s), H2(s), Htot(s), the transfer functions of T1, T2 and T? What is the differentialequation describing the total system T?

(b) Let T be the sum of T1 and T2, T [x(t)] = T2[x(t)] + T1[x(t)], as shown in the diagram.

- SYS 2

- SYS 1

l+ - y(t)

What is Htot(s), the transfer function of the total system T? What is the differential equationdescribing the total system T?

2.4.11 Find the differential equation describing each of the systems of Problem 5 on page 47.

2.4.12 Given a causal LTI system described by the differential equation find H(s), the ROC of H(s), and theimpulse response h(t) of the system. Classify the system as stable/unstable. List the poles of H(s).You should the Matlab residue command for this problem.

(a) y′′′ + 3 y′′ + 2 y′ = x′′ + 6 x′ + 6 x

(b) y′′′ + 8 y′′ + 46 y′ + 68y = 10x′′ + 53 x′ + 144 x

2.4.13 Given a causal LTI system described by

y′(t) +13

y(t) = 2 x(t)

find H(s). Given the input x(t) = e−2tu(t), find the output y(t) without explicitly finding h(t). (UseY (s) = H(s)X(s), and find y(t) from Y (s).)

2.5 Frequency Response

2.5.1 The frequency response of an continuous-time LTI system is given by

Hf (ω) ={

1 for |ω| < 4 π0 for |ω| ≥ 4 π.

Find the output y(t) of the system when the input is

x(t) = 3 cos(2 π t) + 6 sin(5 π t).

2.5.2 The signal x(t) in the previous problem is filtered with a continuous-time LTI system having thefollowing frequency response. Find the output y(t).

−4 0

Hf (ω)

2.5.3 Consider the cascade combination of two continuous-time LTI systems.

x(t) - SYS 1 - SYS 2 - y(t)

The frequency response of SYS 1 is

Hf1 (ω) =

{1 for |ω| < 6 π0 for |ω| ≥ 6 π.

The frequency response of SYS 2 is

Hf2 (ω) =

{0 for |ω| < 4 π1 for |ω| ≥ 4 π.

(a) If the input signal is

x(t) = 2 cos(3π t)− 3 sin(5π t) + 4 cos(7π t)

what is the output signal y(t)?

(b) What is the frequency response of the total system?

2.5.4 The frequency response Hf (ω) of a continuous-time LTI system is given by

Hf (ω) =1

(a) Find the output y(t) when the input is given by

x(t) = cos(2 t) + sin(4 t).

(b) Sketch the magnitude of the frequency response, |Hf (ω)|, and the phase of the frequency response,∠Hf (ω).

(c) Classify the system as stable/unstable and give a brief explanation for your answer.

h(t) = δ(t)− e−t u(t).

(a) What is the frequency response Hf (ω) of this system?

(b) Find and sketch |Hf (ω)|.(c) Is this a lowpass, bandpass, or highpass filter, or none of those?

h(t) = δ(t− 2).

(This is a delay of 2.)

(a) What is the frequency response Hf (ω) of this system?

(b) Find and sketch |Hf (ω)|.(c) Is this a lowpass, bandpass, or highpass filter, or none of those?

2.5.7 When the continuous-time signal x(t)

x(t) = sin(10π t) + cos(5π t)

is applied as the input to an unknown continuous-time system, the observed output signal y(t) is

y(t) = 3 sin(10π t) + 2 cos(7π t).

Which of the following statements is true? Given an explanation for your choice.

2.5.8 The continuous-time signal

x(t) = sin(10π t) u(t) + cos(5π t) u(t)

is sent through a continuous-time LTI system with the frequency response

Hf (ω) = 3− e−j 2.3 ω.

What is the output signal y(t)?

2.5.9 Consider the cascade connection of two copies of the same continuous-time LTI system:

x(t) - h1(t) - h1(t) - y(t)

where h1(t) is as shown:

h1(t)-

−1 0 1 2 3

(a) Find the frequency response Hf1 (ω).

(b) Find the frequency response of the total system Hftot(ω).

2.5.10 Suppose H is an ideal low-pass filter with cut-off frequency ωo. Suppose H is connected in series withanother copy of itself,

x(t) −→ H −→ H −→ y(t)

Find and sketch the frequency response of the total system.

2.5.11 An echo can be modeled with a causal LTI system described by the equation

y(t) = x(t)− 12

x(t− 10).

(a) Find the impulse response h(t).

(b) Classify the system as stable/unstable.

(c) Find the frequency response Hf (ω) and sketch |Hf (ω)|2.

2.5.12 Two continuous-time LTI systems are connected in cascade.

x(t) - h1(t) - h2(t) - y(t)

The impulse responses are given by

h1(t) = δ(t− 0.5)

h2(t) = e−2t u(t)

(a) Find the impulse response h(t) of the total system.

(b) Find the frequency response Hf (ω) of the total system.

(c) Find the steady-state output signal y(t) when the input is

x(t) = cos(2 π t) u(t).

2.6 FourierTransform

For each of the following problems, you are encouraged to use a table of Fourier transforms and properties.

2.6.1 Sketch each of the following signals, and find its Fourier transform. You do not need to perform anyintegration — instead, use the properties of the Fourier transform.

(a) x(t) = e−(t−1) u(t− 1)

(b) x(t) = e−(t−1) u(t)

(c) x(t) = e−t u(t− 1)

2.6.2 Sketch each of the following signals, and find its Fourier transform.

(a) x(t) = e−|t|

(b) x(t) = e−|t−1|

2.6.3 The Fourier transform of x(t) is Xf (ω) = rect(ω).

rect(ω) ={

1 |ω| ≤ 0.50 |ω| > 0.5

Use the Fourier transform properties to sketch the magnitude and phase of each of the following signals.

(a) f(t) = x(t− 2)

(b) g(t) = x(2 t)

(c) v(t) = x(2 t− 1)

(d) q(t) = x(t) ∗ x(t)

2.6.4 Find and sketch the Fourier transform of each of the following signals.

(a) x(t) = cos(6 π t)

(b) x(t) = sinc(3 t)

(c) x(t) = cos(6 π t) sinc(3 t)

(d) x(t) = cos(3 π t) cos(2 π t)

2.6.5 The spectrum Xf (ω) of a continuous-time signal x(t) is given by

Xf (ω) ={

1− |ω|/(3 π) for |ω| ≤ 3 π0 for other values of ω.

Sketch the magnitude and the phase of the Fourier transform of each of the following signals.

a(t) = 3 x(2 t)b(t) = x(t) ∗ x(t)d(t) = x(t) cos(5 πt)f(t) = x(t− 5)

2.6.6 Let the signal g(t) be

g(t) = sinc(2 t).

(a) Find the Fourier transform of g(t).

F{g(t)} = ?

(b) Find the Fourier transform of g(t) convolved with itself.

F{g(t) ∗ g(t)} = ?

(c) Find the Fourier transform of g(t− 1) ∗ g(t− 2).

F{g(t− 1) ∗ g(t− 2)} = ?

(Use part (a) together with properties of the Fourier transform.)

(d) Find the Fourier transform of g(t) multiplied with itself.

F{g(t) · g(t)} = ?

2.6.7 The signal x(t) is a cosine pulse,

x(t) = cos(10π t) [u(t + 1)− u(t− 1)].

Find and make a rough sketch of its Fourier transform Xf (ω). Also, make a sketch of x(t).

2.6.8 Two continuous-time LTI system are used in cascade. Their impulse responses are

h1(t) = sinc(3 t) h2(t) = sinc(5 t).

- h1(t) - h2(t) -

Find the impulse response and sketch the frequency response of the total system.

2.6.9 The signal x(t) is given the product of two sine functions,

x(t) = sin(π t) · sin(2π t).

Find the Fourier transform Xf (ω).

2.6.10 The left-hand column below shows four continuous-time signals. The Fourier transform of each signalappears in the right-hand column in mixed-up order. Match the signal to its Fourier transform.

SIGNAL FOURIER TRANSFORM

−5 0 50

−5 0 5−1

−0.5

Xf (ω

−5 0 50

−5 0 5

Xf (ω

−5 0 5−0.5

−5 0 50

Xf (ω

−5 0 5−1

−0.5

−5 0 50

Xf (ω

2.6.11 The ideal continuous-time band-stop filter has the frequency response

Hf (ω) =

1 |ω| ≤ ω1

0 ω1 < |ω| < ω2

1 |ω| ≥ ω2.

(a) What is the impulse response h(t) of the ideal band-stop filter?

(b) Describe how to implement the ideal band-stop filter using only lowpass and highpass filters.

2.6.12 The ideal continuous-time band-pass filter has the frequency response

Hf (ω) =

0 |ω| ≤ ω1

1 ω1 < |ω| < ω2

0 |ω| ≥ ω2.

Describe how to implement the ideal band-pass using two ideal low-pass filters with different cut-offfrequencies. (Should a parallel or cascade structure be used?) Using that, find the impulse responseh(t) of the ideal band-pass filter.

2.6.13 The ideal continuous-time high-pass filter has the frequency response

Hf (ω) ={

0, |ω| ≤ ωc

1, |ω| > ωc.

Find the impulse response h(t).

2.6.14 The frequency response of a continuous-time LTI is given by

Hf (ω) ={

0 |ω| < 2 π1 |ω| ≥ 2 π

(This is an ideal high-pass filter.) Use the Fourier transform to find the output y(t) when the inputx(t) is given by

(a) x(t) = sinc(t) = sin(π t)π t

(b) x(t) = sinc(3 t) = sin(3 π t)3 π t

2.6.15 What is the Fourier transform of x(t)?

x(t) = cos(0.3 π t) · sin(0.1 π t)

Xf (ω) = F{x(t)} = ?

2.7 Matching

2.7.1 Match the impulse response h(t) of a continuous-time LTI system with the correct plot of its frequencyresponse |Hf (ω)|. Explain how you obtain your answer.

−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−0.5

IMPULSE RESPONSE

−8 −6 −4 −2 0 2 4 6 80

2.7.2 For the impulse response h(t) illustrated in the previous problem, identify the correct diagram of thepoles of H(s). Explain how you obtain your answer.

−3 −2 −1 0 1 2 3−10

Real part

−3 −2 −1 0 1 2 3−10

Real part

−3 −2 −1 0 1 2 3−10

Real part

−3 −2 −1 0 1 2 3−10

Real part

2.7.3 The following diagrams indicate the pole locations of six continuous-time LTI systems. Match eachwith the corresponding impulse response with out actually computing the Laplace transform.

POLE-ZERO DIAGRAM IMPULSE RESPONSE

123456

−1 0 1−6

2.7.4 The figure shows the impulse responses and frequency responses of four continuous-time LTI systems.But they are out of order. Match the impulse response to its frequency response magnitude, andexplain your answer.

IMPULSE RESPONSE FREQUENCY RESPONSE

−1 0 1 2 3 4 5 6−1

−0.5

1IMPULSE RESPONSE 1

−10 −5 0 5 100

0.6FREQUENCY RESPONSE 1

−1 0 1 2 3 4 5 6−1

−0.5

1IMPULSE RESPONSE 2

−10 −5 0 5 100

−1 0 1 2 3 4 5 6−1

−0.5

1IMPULSE RESPONSE 3

−10 −5 0 5 100

−1 0 1 2 3 4 5 6−1

−0.5

1IMPULSE RESPONSE 4

TIME (SEC)−10 −5 0 5 100

FREQUENCY (Hz)

2.7.5 The figure shows the impulse responses and frequency responses of four continuous-time LTI systems.But they are out of order. Match each impulse response with its frequency response.

IMPULSE RESPONSE FREQUENCY RESPONSE

−1 0 1 2 3 4 5 6

−0.5

IMPULSE RESPONSE 1

−10 −5 0 5 10

−1 0 1 2 3 4 5 6

−0.5

IMPULSE RESPONSE 2

−10 −5 0 5 10

FREQUENCY (Hz)

−1 0 1 2 3 4 5 6

−0.5

IMPULSE RESPONSE 3

−10 −5 0 5 10

−1 0 1 2 3 4 5 6

−0.5

IMPULSE RESPONSE 4

TIME (SEC)

−10 −5 0 5 10

2.7.6 The figure shows the pole diagrams and frequency responses of four continuous-time LTI systems. Butthey are out of order. Match each pole diagram with its frequency response.

POLE DIAGRAM FREQUENCY RESPONSE

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5−20

20POLE−ZERO DIAGRAM 1

−6 −4 −2 0 2 4 6

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5−20

−6 −4 −2 0 2 4 6

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5−20

−6 −4 −2 0 2 4 6

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5−20

−6 −4 −2 0 2 4 6

FREQUENCY (Hz)

2.7.7 The diagrams on the following pages show the pole-zero diagrams and frequency responses of 8 causalcontinuous-time LTI systems. But the diagrams are out of order. Match each diagram by filling outthe following table.

In the pole-zero diagrams, the zeros are shown with ‘o’ and the poles are shown by ×.

POLE-ZERO DIAGRAM FREQUENCY RESPONSE

12345678

−6 −4 −2 0 2 4 6

−20 −10 0 10 200

1FREQUENCY RESPONSE 4

−20 −10 0 10 200

2.7.8 The figures on the next pages show a continuous-time signal x(t) and its spectrum. This signal consistsof one cosine pulse followed by another. This signal is used as the input to four continuous-time LTIsystems (Systems A, B, C, and D) and four output signals are obtained. The frequency responses ofeach of these four systems are also shown below. The last figure shows the four output signals. Butthe output signals are out of order. For each of the four systems, identify which signal is the outputsignal by completing the table below. Explain your answer.

System Output SignalABCD

0 1 2 3 4 5 60

SPECTRUM of INPUT SIGNAL

FREQENCY (HERZ)

0 1 2 3 4 5 60

FREQUENCY RESPONSE

0 1 2 3 4 5 60

FREQUENCY RESPONSE

0 1 2 3 4 5 60

FREQUENCY (HERZ)

FREQUENCY RESPONSE

0 1 2 3 4 5 60

FREQUENCY (HERZ)

FREQUENCY RESPONSE

0 1 2 3 4 5 6 7 8 9 10

−0.5

TIME (SECONDS)

INPUT SIGNAL

0 1 2 3 4 5 6 7 8 9 10−1.5

−0.5

OUTPUT SIGNALS

0 1 2 3 4 5 6 7 8 9 10−1.5

−0.5

0 1 2 3 4 5 6 7 8 9 10−1.5

−0.5

0 1 2 3 4 5 6 7 8 9 10−1.5

−0.5

TIME (SECONDS)

2.7.9 The figure shows two input signals x1(t) and x2(t), two impulse responses h1(t) and h2(t), and fouroutput signals y1(t), y2(t), y3(t), y4(t). Identify which input signal and which impulse response causeseach of the four output signals.

(Your answer should have four parts, y1(t) = h?(t) ∗ x?(t), etc.)

2.7.10 The figure shows two input signals x1(t) and x2(t), two impulse responses h1(t) and h2(t), and fouroutput signals y1(t), y2(t), y3(t), y4(t). Identify which input signal and which impulse response causeseach of the four output signals.

(Your answer should have four parts, y1(t) = h?(t) ∗ x?(t), etc.)

2.8 FourierSeries

2.8.1 Find the Fourier series coefficients c(k) for the periodic signal

x(t) = | cos(t)|.

2.8.2 The periodic signal x(t) is given by

x(t) = | sin(t)|.

(a) Sketch the signal x(t). What is its period?

(b) Find the Fourier series coefficients of x(t). Simplify!

(c) Is x(t) band-limited?

2.8.3 Consider the signal

x(t) = cos(10π t)

(a) Write the Fourier series expansion of x(t).

(b) Sketch the line spectrum of x(t).

x(t) = cos(12π t) cos(8 π t).

(a) Write the Fourier series expansion of x(t).

(b) Sketch the line spectrum of x(t).

2.8.5 Find the Fourier series coefficients c(k) for the periodic triangular pulse shown below.

−5 −4 −3 −2 −1 0 1 2 3 4 5

· · ·�

��@@@�

��@@@

· · ·

2.8.6 The signal x(t) is

x(t) = [cos(π t)]2 sin(πt).

(a) Find its Fourier transform Xf (ω).

(b) Is x(t) periodic? If so, determine the Fourier series coefficients of x(t).

2.8.7 The signal x(t), which is periodic with period T = 1/4, has the Fourier series coefficients

1 + k2.

The signal is filtered with an ideal lowpass filter with cut-off frequency of fc = 5.2 Hz. What is theoutput of the filter?

Simplify your answer so it does not contain any complex numbers. Note: ω = 2π f converts fromphysical frequency to angular frequency.

x(t) = |0.5 + cos(4πt)|

The signal is filtered with an ideal bandpass filter that only passes frequencies between 2.5 Hz and 3.5Hz.

Hf (ω) =

0 |ω| < 5 π

1 5 π ≤ |ω| ≤ 7 π

0 |ω| > 7 π

(a) Sketch the input signal x(t).

(b) Find and sketch the output signal y(t). Hint: Use the Fourier series, but do not compute theFourier series of x(t).

2.9 Modulation

2.9.1 Suppose the signal x(t) has the following spectrum X(ω).

2π 3πω

Suppose the signal x(t) is sent through the following continuous-time system.

x(t) - g× -a(t)

h(t) - y(t)6

cos(3π t)

where the frequency response H(ω) is an ideal low-pass,

�−2 π 0

2 πω

(a) Sketch the spectrums (Fourier transforms) of cos(3 π t), a(t), and y(t).

(b) Is the total system an LTI system?

2.9.2 Suppose h(t) is the impulse response of a lowpass filter with frequency response shown.

-−π 0

Suppose a new system is constructed so that its impulse response h2(t) is given by

h2(t) = 2 cos(6 π t) · h(t).

(a) Sketch the frequency response of the new system.

(b) Is the new system a low-pass, band-pass, or high-pass filter?

(c) If the signal

x(t) = 2 cos (3π t) + 4 sin (6π t)

is sent through the new system, what is the output y(t)?

x(t) - h2(t) - y(t)

2.10 Summary

2.10.1 Consider the parallel combination of two continuous-time LTI systems.

- h2(t)

- h1(t)

j+ - y(t)

You are told that the step response of the upper branch is

s1(t) = 2 u(t) + u(t− 1).

You observe that the impulse response of the total system is

h(t) = 2 δ(t) + δ(t− 2).

Find and sketch h1(t) and h2(t).

2.10.2 Lets define two continuous-time signals,

x1(t) = δ(t) + 2 δ(t− 1) + δ(t− 2),

x2(t) = δ(t) + 3 δ(t− 1) + 3 δ(t− 2) + δ(t− 3).

(a) It turns out x2(t) can be written as x2(t) = x1(t) ∗ f(t). What is f(t)?

(b) When x1(t) is applied to some continuous-time LTI system, the following output signal y1(t) isobserved.

−1 0 1 2 3 4 5

y1(t)2

��HHH

Sketch the output signal y2(t) that results when x2(t) is applied to the same system.

2.10.3 When the signal x(t)

x(t) = sin(10π t)

is applied as the input to an unknown continuous-time system, the observed output signal y(t) is

y(t) = u(t)− u(t− 2).

Which of the following statements is true? Given an explanation for your choice.

2.10.4 The output y(t) of a continuous-time LTI system is given by

y(t) = 3 x(t)− x(t− 1.5) + x(t− 2)

where x(t) is the input to the system.

(a) What is the impulse response h(t) of this system?

(b) What is the transfer function H(s) of this system?

2.10.5

2.10.6 Two continuous-time LTI systems have the following impulse responses

h1(t) = e−2 t u(t) h2(t) = e−3 t u(t)

(a) If the two systems are connected in parallel,

- h2(t)

- h1(t)

j+ - y(t)

What is the frequency response of the total system?What differential equation describes this system?

(b) If the two systems are connected in series,

x(t) - h1(t) - h2(t) - y(t)

What is the frequency response of the total system?What differential equation describes this system?

2.10.7 The impulse response of a continuous-time LTI is given by

h(t) = u(t)− u(t− 2).

Find the output y(t) when the input x(t) is given by

x(t) = t u(t).

Classify the system as BIBO stable/unstable.

2.10.8 Consider the cascade of two causal continuous-time LTI systems.

- H1(s) - H2(s) -

You are told that the first system is described by the differential equation

y′ + 2 y = x.

You observe that the step response of the total system is

s(t) = e−2t u(t)− 2 e−t u(t) + u(t).

Find and sketch the impulse response of the second system.

2.10.9 Consider the following connection of two continuous-time LTI systems.

h1(t)6

You are told that

h1(t) = e−t u(t), h2(t) = −2 e−4t u(t).

Find the impulse response h(t) of the total system.

2.10.10 The frequency response of a continuous-time LTI system is given by

Hf (ω) = 2 e−jω.

(a) Find and sketch the impulse response h(t).

(b) Sketch the output of this system when the input is

x(t) =∞∑

k=−∞

δ(t− 4 k).

2.10.11 Suppose the impulse response of a continuous-time LTI system has the form

h(t) = B e−3t sin(5π t) u(t).

Suppose a signal x(t) with the form

x(t) = A cos(10π t− 0.5 π) u(t)

is applied to the system. What is the generic form of the output signal?

y′′(t) + 5 y′(t) + 4 y(t) = 3 x′(t) + 2 x(t).

(a) Find the transfer function H(s) and its region of convergence, and sketch the pole-zero diagramfor this system.

(b) When the input signal is

x(t) = 3 e−t cos(2 t) u(t)

find the ‘generic’ form of the output signal y(t). (I mean that you do not have to compute theresidue of the partial fraction expansion; you may leave them as A, B, etc.)

h(t) = e−3 t sin(2t) u(t).

Suppose the input signal is given by

x(t) = e−t u(t).

(a) Without doing any arithmetic, write down the form you know the output signal y(t) must take.

(b) Find the transfer function.

(c) Find the differential equation that represents this system.

(d) Find the frequency response Hf (ω) of this system.

(e) Sketch the pole diagram of the system.

2.10.14 A continuous-time LTI system has the following impulse response:

−1 0 1 2 3

The signal

x(t) = 4 + 3 cos(2π t)

is sent through the system.

x(t) - h(t) - y(t)

(a) Sketch x(t).

(b) Find and sketch the output y(t).

2.10.15 Consider the cascade connection of two continuous-time LTI systems

x(t) - SYS 1 - SYS 2 - y(t)

with impulse responses

h1(t) = 4 e−2t cos(5π t) u(t)

h2(t) = 5 e−3t u(t)

(a) Sketch the pole diagram of the two transfer functions H1(s), H2(s),

(b) Sketch the pole diagram of the transfer function of the total system Htot(s).

(c) Which form can the total impulse response htot(t) take? (Choose all that apply.)

1) A e−2t u(t) + B e−3t u(t) + C cos(5π t + θ) u(t)2) A e−5t cos(5π t + θ) u(t)3) A e−3t cos(5π t + θ) u(t) + B e−2t u(t)4) A e−2t cos(5π t + θ) u(t) + B e−3t u(t)5) A e−2t cos(5π t + θ1) u(t) + B e−3t cos(5π t + θ2)u(t)6) A e−2t cos(5π t + θ1) u(t) + B e−3t cos(5π t + θ2)u(t) + C e−2t u(t) + D e−3t u(t)

2.10.16 Suppose the impulse response of a continuous-time LTI system is

h(t) = 3 e−t u(t)

and the signal

x(t) = 2 cos(t) u(t)

is applied to the system. Find the steady-state output. (First find Hf (ω).)

2.10.17 The frequency response of a continuous-time LTI filter is given by

Hf (ω) =

2, |ω| ≤ 2 π1, 2 π < |ω| ≤ 4 π0, 4 π < |ω|.

Hf (ω)

−4π −2π 2π 4π

Find the impulse response h(t).

2.10.18 Suppose a tape recording is made with an inferior microphone which has the frequency response

Hf (ω) ={

1− |ω|W , |ω| < W

0, |ω| ≥ W,

where W = 1200π (600 Hz).

(a) What is the frequency response of the system you would use to compensate for the distortioncaused by the microphone? Make a sketch of your frequency response and explain your answer.

(b) Suppose you wish to digitally store a recording that was made with this microphone. Whatsampling rate would you use? Give your answer in Herz.

2.10.19 A continuous-time LTI system H(s) is described by the differential equation

y′′ + 2 y′ + y = x.

You are considering the design of an LTI system G(s) that inverts H(s).

(a) Find the frequency response Gf (ω) of the inverse system.

(b) Find and sketch |Gf (ω)|.(c) Bonus: Why will it be difficult to implement G(s) exactly?

2.10.20 Which of the following diagrams shows the frequency response of an elliptic filter? (Recall the Matlabexercise on filter design.)

−5 0 50

|H1f (ω)|

−5 0 50

|H2f (ω)|

−5 0 50

|H3f (ω)|

Also identify the Chebyshev-I filter and the Butterworth filter.

3 The Sampling Theorem

3.0.1 The Fourier transform of a signal x(t) satisfies

X(ω) = 0 for |ω| ≥ 8 π.

What is the maximum sampling period T such that x(t) can be recovered from the samples x(nT )?

3.0.2 For the following continuous-time system

x(t) - g× -a(t)h(t) - y(t)

δs(t)

the input to the filter h(t) is the product of x(t) and δs(t):

a(t) = x(t) · δs(t)

and the impulse train is given by

δs(t) =∞∑

n=−∞δ(t− n Ts)

where the sampling period is

Ts =23.

The frequency response H(ω) is an ideal low-pass,

−2 π 0

2 πω

Suppose the input signal is

x(t) = 4 cos(6 π t).

(a) Sketch the spectrums (Fourier transforms) of x(t), δs(t), a(t), and y(t).

(b) Find and sketch a(t) and y(t).

(c) Does y(t) = x(t)? Does this contradict the sampling theorem? Explain.

3.0.3 For the following continuous-time system

x(t) - g× -a(t)

h(t) - y(t)6

δT (t)

the input to the filter h(t) is the product of x(t) and δT (t):

a(t) = x(t) · δT (t)

and the impulse train is given by

δT (t) =∞∑

n=−∞δ(t− n T ).

The filter h(t) is not an ideal low-pass filter. Instead, its frequency response H(ω) is

H(ω)T

−12π −8π −4π 0 4π 8π 12π

−6π 6π

� �� A

What is the slowest sampling rate such that y(t) = x(t) when . . .

(a) the spectrum of the signal x(t) satisfies

X(ω) = 0 for |ω| ≥ 6 π.

(b) when the spectrum of the signal x(t) satisfies

X(ω) = 0 for |ω| ≥ 4 π.

Express the sampling rate in samples/sec.

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Signals and Systems - Rice University1 1 Cover Page.1.1 Signals and Systems: Elec 301 summary: This course deals with signals, systems, and transforms, from their theoretical mathematical