Signals: The Basics

Question 1

what is the mathematical definition of a signal?

Answer 1

function of one/more independent variables

Question 2

what is the notation for continuous independent variables?

Answer 2

(.)

Question 3

what is the notation for discrete independent variables?

Answer 3

[.]

Question 4

define: x[n]

Answer 4

discrete signal (could be in time)

Question 5

define: x(t)

Answer 5

continuous signal in time

Question 6

define: x(nT)

Answer 6

a continuous signal in time that is sampled at nT

Question 7

x[n] = ________ = x(nT)

Answer 7

x(t)|t=nT

Question 8

define: x(t)|t=nT

Answer 8

continuous sample x(t) evaluated at t = nT

Question 9

what are examples of x(t)|t=nT ?

Answer 9

• analog to digital converter

- digital camera

Question 10

periodicity of continuous signals?

Answer 10

x(t + To) = x(t) for all t

To = period, which is usually time

Question 11

periodicity of discrete signals?

Answer 11

x[n + N] = x[n] for all n

N = period, which is usually # of samples

Question 12

x(t +To) =

Answer 12

x(t) for all t

Question 13

x[n + N] =

Answer 13

x[n] for all n

Question 14

what is the fundamental period of x[n] = sin^2(pi*n/5)

Answer 14

5

Question 15

how would you determine the fundamental period of x[n] = sin^2(pi*n/5)

Answer 15

x[n] = sin^2(pin/5)
= sin^2(pin/5 + pi)
= sin^2(pi*(n+5)/5)
= x[n+5]

Question 16

what is the fundamental period of x[n] = cos^2(pi*n/6)

Answer 16

6

Question 17

how would you determine the fundamental period of x[n] = cos^2(pi*n/6)

Answer 17

x[n] = cos^2(pin/6)
= 0.5+0.5cos(2pin/6)
= 0.5+0.5cos(2pin/6 + 2pi) [2pi = smallest phase
shift that doesn’t change
cosine]
= 0.5+0.5cos(2pi(n+6)/6)
= cos^2(pi(n+6)/6)
= x[n+6]

Question 18

x(t) = Acos(2pif*t + theta)

given A, f, and theta how would you determine the fundamental period?

Answer 18

x(t) = Acos(2pift + theta)
= Acos(2pift + theta + 2pi)
= Acos(2pi*f(t + 1/f) + theta)
= x(t + 1/f)

so fundamental period T = 1/f