1

Question 1

Whats a signal?

Answer 1

-Functions of 1 or more independent variables that carry information

Question 2

Whats a system?

Answer 2

-A system processes a signal and has inputs and outputs.
We can have a input being continuous or discrete.
- Systems can be linear or nonlinear
-Time invariant or time variant

Question 3

1D- Example signals:

Answer 3

• speech vs amplitude graph
• stock price vs price
• Electrocardiogram (ECG) trace vs voltage

Question 4

2D- Example signals:

Answer 4

• photo
• Measured MRI data
• A powerpoint slide

Question 5

System examples:

Answer 5

• Microphone
• MRI scanner
• Webcam

Question 6

Example 2D discrete function:

Answer 6

-A slice of a MR image

Question 7

A 3D function gives one value for three independent variables

Answer 7

-3 orthogonal slices through a 3D brain image

Question 8

Example 3D function

Answer 8

-A video

Question 9

continuous wave function:

Answer 9

-x(t) = Acos(w(0)+phi)
A - amplitude
w(0) - frequency
phi - phase

Question 10

Whats does a time shift cause?

Answer 10

A phase change

Question 11

Whats does a time shift cause?

Answer 11

A phase change

-Within this there’s always a value of t(0) that equivalents to a phase shift

Question 12

What does changing the phase equate to?

Answer 12

• Changing the time variable

Question 13

When is a function even?
Periodic function:
x(t) = x(t+T(0))
even: x(t) = x(-t)

Answer 13

• When its reflected about the origin, its exactly the same, its symmetric, reflected in the y axis
• If we replace the time argument by its negative, the function itself doesn’t change.

Question 14

When is a function odd?

Whats the algebraic expression?

Answer 14

• The mirror image flipped over
• x(t) = x(t+T(0))
  odd: x(t) = -x(t)

Question 15

-What do we know about discrete time cases?

Answer 15

In discrete time the functions are not periodic

Question 16

How do we know if a function is periodic?

Answer 16

• If omega(0)(the period) is a multiple of 2*pi

- If not then its not periodic

Question 17

What do we know about time and space change for a continuous function?

Answer 17

• They are always equivalent
• Is always periodic
• Its periodic for any case of omega (0)

Question 18

In the discrete time case, how do we know if its continuous?

Answer 18

-Its periodic only if 2*pi/omega(0) can be multiplied by an integer to get another integer

Question 19

In the discrete time case, as we vary the frequency (omega(0)), what happens?

Answer 19

• We only see distinct signals for omega(0) varying over a 2*pi interval. we’ll see the same sequences repeat.
• sinusoidal functions will be the same

Question 20

In the continuous time case, as we vary frequency?

Answer 20

we’ll always see different sinusoidal signals.

-sinusoidal functions will always be different

Question 21

For an exponential, what does a time shift correspond to?

Answer 21

• A scale change

- Ce^(at)

Question 22

Real exponential: discrete time case

Answer 22

Ce^(Beta(n)) = C(alpha)^n

Question 23

Whats -j equal to?

Answer 23

e^(-jpi/2)

Question 24

Whats 1 equal to?

Answer 24

e^(jalpha/2) * e^(-jalpha/2)

Question 25

What can we rewrite 2jsin(-x) as?

Answer 25

(-2jsin(x))
or
e^(-x) - e^(x)

Question 26

What is the frequency and time period independent of?

Answer 26

-The time delay and phase difference

Question 27

How may we know if two signals are the same for all values of t? This is for discrete time signals

Answer 27

y(t) = x(t) for all values of t if:
w(y)=w(x) 
and
w(x)T(x) + theta(x) = w(y)T(y)+theta(y)+2*pi*k
-Remember k can equal 0

Question 28

How to find the period for a discreet time signal?

Answer 28

We need the smallest N such that omegaN = 2pik for some integer k > 0.
-If we get an integer for n then the function is periodic
pi/3N = 2pik
T = 6
k = (pi/3)6=2pi so k =1

Question 29

Re write the discrete signal x[4-n]?

Answer 29

x[-(n-4)]

Question 30

What does x[2n] do?

Answer 30

Generates a new signal with x[n] for even values of n

Question 31

What difficulty arises when we try to define a signal as.x[n/2]?

Answer 31

The difficulty arises when we try to evaluate x[n/2] at n = 1, for example (or generally for n an odd integer). Since x[1/2] is not defined, the signal x[n/2] does not exist.

Question 32

cos(t) in exponential form?

Answer 32

e^(t)+e(t)/2

Question 33

sin(t) in exponential form?

Answer 33

e^(jt)-e^(-jt)/2j

Question 34

Let x[n] and y[n] be periodic signals with fundamental periods Ni and N2, respectively. Under what conditions is the sum x[n] + y[n] periodic, and what is the fundamental period of this signal if it is periodic?

Answer 34

Similarly, x[n] + y[n] will be periodic if there exist integers n and k such that nN = kN2. But such integers always exist, a trivial example being n = N2 and k = N1 . So the sum is always periodic with period nN, for n the smallest allow­ able integer.

Question 35

Let x(t) and y(t) be periodic signals with fundamental periods Ti and T2, respec­ tively. Under what conditions is the sum x(t) + y(t) periodic, and what is the fundamental period of this signal if it is periodic?

Answer 35

The sum x(t) + y(t) will be periodic if there exist integers n and k such that nT1 = kT 2, that is, if x(t) and y(t) have a common (possibly not fundamental) period. The fundamental period of the combined signal will be nT1 for the small­ est allowable n.