1 Flashcards
Whats a signal?
-Functions of 1 or more independent variables that carry information
Whats a system?
-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
1D- Example signals:
- speech vs amplitude graph
- stock price vs price
- Electrocardiogram (ECG) trace vs voltage
2D- Example signals:
- photo
- Measured MRI data
- A powerpoint slide
System examples:
- Microphone
- MRI scanner
- Webcam
Example 2D discrete function:
-A slice of a MR image
A 3D function gives one value for three independent variables
-3 orthogonal slices through a 3D brain image
Example 3D function
-A video
continuous wave function:
-x(t) = Acos(w(0)+phi)
A - amplitude
w(0) - frequency
phi - phase
Whats does a time shift cause?
A phase change
Whats does a time shift cause?
A phase change
-Within this there’s always a value of t(0) that equivalents to a phase shift
What does changing the phase equate to?
- Changing the time variable
When is a function even?
Periodic function:
x(t) = x(t+T(0))
even: x(t) = x(-t)
- 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.
When is a function odd?
Whats the algebraic expression?
- The mirror image flipped over
- x(t) = x(t+T(0))
odd: x(t) = -x(t)
-What do we know about discrete time cases?
In discrete time the functions are not periodic
How do we know if a function is periodic?
- If omega(0)(the period) is a multiple of 2*pi
- If not then its not periodic
What do we know about time and space change for a continuous function?
- They are always equivalent
- Is always periodic
- Its periodic for any case of omega (0)
In the discrete time case, how do we know if its continuous?
-Its periodic only if 2*pi/omega(0) can be multiplied by an integer to get another integer
In the discrete time case, as we vary the frequency (omega(0)), what happens?
- 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
In the continuous time case, as we vary frequency?
we’ll always see different sinusoidal signals.
-sinusoidal functions will always be different
For an exponential, what does a time shift correspond to?
- A scale change
- Ce^(at)
Real exponential: discrete time case
Ce^(Beta(n)) = C(alpha)^n
Whats -j equal to?
e^(-jpi/2)
Whats 1 equal to?
e^(jalpha/2) * e^(-jalpha/2)
What can we rewrite 2jsin(-x) as?
(-2jsin(x))
or
e^(-x) - e^(x)
What is the frequency and time period independent of?
-The time delay and phase difference
How may we know if two signals are the same for all values of t? This is for discrete time signals
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
How to find the period for a discreet time signal?
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
Re write the discrete signal x[4-n]?
x[-(n-4)]
What does x[2n] do?
Generates a new signal with x[n] for even values of n
What difficulty arises when we try to define a signal as.x[n/2]?
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.
cos(t) in exponential form?
e^(t)+e(t)/2
sin(t) in exponential form?
e^(jt)-e^(-jt)/2j
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?
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.
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?
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.
