Signal Fundamentals Flashcards
Signal Classifications:
Pairs/Axes (7)
Continuous or Discrete
Analog or Digital
Periodic or Aperiodic
Deterministic or Random/Probabilistic
Energy or Power
Causal or Non-Causal
Even or Odd
What is a Signal?
A set of data or information.
Typically represented as a function of the independent variable time, but not always.
Can include anything that ‘transmits’ information, energy, power:
Examples:
- Telephone signal
- A companies’ monthly sales
- Power delivery from an engine to wheels
- Stock prices
Signal Classifications:
Analog Signals
Analog signals have a signal amplituded
which can take an infinite number of possible values, bounded by some range.
Signal Classifications:
Digital Signals
In a digital signal, the amplitude is restricted to a set of distinct possible values.
The most common digital signal is binary, taking only the values 0 or 1
Signal Classifications:
Periodic Signals
A signal is periodic if it repeats in constant intervals.
Formally:
For some positive constant, T0 - Called the period,
x(t) = x(t + T0) for all t
The signal repeats with a period of T0
The smallest value of T0 that satisfies this is called the Fundamental Period of the signal.
Signal Classifications:
What is the Fundamental Period
of a Periodic Signal?
The smallest change of time that the signal repeats itself in.
Formally, the smallest value T0 that satisfies:
x(t) = x( t + T0 ) for all t
Signal Classifications:
Aperiodic Signal
A signal that is not periodic.
It either has no repeating values, such as an exponential function, or it’s values do not repeat in a regular fashion.
Signal Classifications:
Analog vs Digital
Analog signals can take infinite values within a range.
Digital signals can only take specific values, usually 0 or 1
Signal Classifications:
Periodic Signals
vs
Aperiodic Signals
Periodic Signals have a repeating pattern.
Aperiodic signals do not.
Signal Classifications:
Continous Time Signal
A signal where there are an infinite number of values within a portion of the domain.
There is a value of x(t) for every value of t in the domain.
Example: t can be 0, 0.1, 0.0000000001, etc
Signal Classifications:
Discrete Time Signals
In Discrete Time Signals,
the domain is separated into discrete steps.
The function is only defined at these steps.
For a function x(t), only some values of t have a corresponding x(t).
The steps are usually very small and spread regularly.
Example: A signal recorded by taking regular samples
Signal Classifications:
Continuous vs Discrete Time Signals
Continous Time Signals
have a value for any possible t in the domain
Discrete Time Signals
are only defined at values of t within a particular set
Signal Classifications:
Energy Signals
Refers to how the “size” of the signal is measured.
An Energy Signal
is one that has finite energy, basically one that “dies out”
Formally:
x(t) → 0 as | t | → infinity
Signal Classifications:
Power Signals
Refers to how the “size” of the signal is measured
A Power Signal
is one that doesn’t “die out” over time
Formally:
x(t) does not approach 0 as | t | → infinity
All Periodic Signals are Power Signals
Trying to measure the Energy would yield an infinite result, so the signal size is measured as “Power”
Signal Classfications:
Energy vs Power Signals
Energy Signals
The amplitude approaches 0 over time, so the signal size can be measured as “Energy”
Power Signals
The amplitude does not approach 0 over time, signal size must be measured as “Power”
Important Note:
A signal belongs in neither classification if it’s Power, Px is calculated to be infinite
Signal Classifications:
Causal
vs
Non-causal
Causal Signals
- Begin at or after t=0
- Do not extend to t= -inf
- Signal amplitude is not affected by “future” values
- Implies that changes are “caused” by something immediate
Non-Causal Signals
- Continuos towards t= -inf
- “Future” values may affect “current” values
Signal Classfications:
Deterministic
vs
Random(Probabilistic)
Deterministic Signals
- The signal x(t) is a function of t
Random or Probabilistic Signals
- The signal x(t) is NOT a function of t
- But it may be related with probabilities
Signal Classification:
Even Signals
vs
Odd Signals
Refers to the type of Symmetry a signal/function may have
Even Signals/Functions
- Reflect across the Y-axis, so the “left” is a mirror image of the “right”
- No sign change
- x(-t) = x(t)
Odd Signals/Functions
- Reflect across the Y-axis AND the X-axis
- Sign change
- x(-t) = - x(t)
Even and Odd Functions:
What is their importance in analyzing signals?
Every signal x(t) can be expressed as a sum of Even and Odd component functions:
x(t) =
(1/2)[x(t) + x(-t)] (Even component)
+ (1/2)[x(t) - x(-t)] (Odd component)
Even and Odd Functions:
Multiplication Rules
Apply when multiplying two functions together
Even X Odd = Odd
Odd X Odd = Even
Even X Even = Even
Note: It sort of parallels the rules of multiplying positive and negative values
Signal Size:
Two ways to measure the
size of a signal
Signal Energy
Used when signal exists for finite time
Signal Power
Used when signal exists for infinite(in effect) time, such as a periodic signal
Signal Size:
Why is signal size NOT just the integral of the signal?
The integral of the signal would calculate the area under the curve.
BUT, in most signals, this includes both negative and positive areas that cancel each other out.
This would show a signal size that is much smaller than the actual signal in many instances.
Signal Size:
What conditions are required for
Energy to be an appropriate measure of
signal size?
Signal energy must be finite
in order to be a useful measure.
So:
The signal amplitude must approach 0 as time approaches infinity and negative infinity:
lim|t|→inf x(t) = 0
If this condition is not met, Power of the signal should be used to measure the signal size
Signal Size:
Energy of a Signal
Formula
The Energy of the Signal, Ex
is the Integral of the square of the amplitude, wrt time
Ex = INT( x(t)2 )dt
Note: More generally, use the absolute value of x(t) for complex valued signals
Signal Size:
Relationship between
Energy and Power
A signal with Finite Energy, Ex = c,
has Zero Power: Px = 0
A signal with Finite Power, Px = c
has Infinite Energy: Ex = INF
Signal Size:
Power of a Signal
Basic Idea
The energy of a signal is infinite. Presumably because it is periodic.
Power is a measure of the time average of the energy over a period.
Signal Size:
Power of a Signal
Formula
For a Real signal:
Px =
limT→∞ (1/T) ∫-T/2T/2 x2(t) dt
Note: If period T is specified, ignore the limit
Complex Signal:
Px =
limT→∞ (1/T) ∫-T/2T/2 |x2(t)| dt
Signal Size:
What is the Root Mean Square (RMS)
of a Signal?
The RMS is simply the square of the Power of the signal.
rms = Px2
Fundamental Signals:
List of Basic Functions/Signals
Unit Step Function, u(t)
Unit Impulse Function, δ(t)
Unit Ramp Signal, ur(t)
Exponential Signal, est
Pulse Signals
Fundamental Signals:
Unit Step Function
u(t) = 1 when t ≥ 0
u(t) = 0 when t < 0
- Amplitude is zero before t=0, then immediately jumps to 1 and stays there
- Useful to describe signals that begin at t=0
- Multiply the function by u(t) to “get rid of” everything before 0, creating a causal form of the signal
Fundamental Signals:
Unit Impulse Function
δ(t)
Represents an “impulse” at exactly zero
δ(t) = 0 when t ≠ 0
and ∫ δ(t)dt = 1
Very useful in determing system responses.
Can be approximated with very short-duration pulse functions, so long as the function integrates to Unity( 1 )
Fundamental Signals:
Impulse Function δ(t)
Some approximation functions
The impulse function can be approximated by several different pulse functions, so long as the area under the pulse function is unity.
- Rectangular Function
- f(t) = 1/ε , -ε/2 < t
- ε → 0
- Exponential
- f(t) = e-𝛼t , t > 0
- 𝛼→∞
- Triangular
- f(t) goes from 0 to 1/ε at -ε < t < 0
- f(t) goes from 1/ε to 0 at 0 < t
- ε → 0
- Guassian
- some bullshit you’ll look up when you need it
Fundamental Signals:
Pulse Signals
A Pulse signal
is a signal that lasts for just a short time.
The simplest way to obtain a pulse signal is to just subtract one step function from another, creating a Rectangular Pulse
Ex:
u(t-2) - u(t-4)
Creates a “rectangular” pulse between t=2 and t=4
Fundamental Signals:
Exponential Function
est
Very important to analysis of signals and systems.
- S is the complex frequency
- s = σ + jω
- ω is the real frequency component
Fundamental Signals:
Exponential Function:
Representing Complex Frequency S
as a complex sinusoid
est
s = σ + jω
est = e(σ + jω)t = eσtejωt
ejωt can be represented as the complex sinusoid:
cos(ωt) + jsin(ωt)
So
est = eσt ( cos(ωt) + jsin(ωt) )
Why: Supports finding the conjugate relation
Fundamental Signals:
Exponential Function:
Conjugate
and
Conjugate Relation
est = e(σ + jω)t = eσtejωt
The Conjugate es*t = e(σ - jω)t = eσt / ejωt
1/ejωt can be represented as the complex sinusoid:
cos(ωt) - jsin(ωt)
So
es*t = eσt ( cos(ωt) - jsin(ωt) )
We can use this to get the Conjugate Relation for the Exponentional Function:
eσtcos(ωt) = (1/2)(est + es*t)
Fundamental Signals:
Sampling Property
of the
Unit Impulse Function
For a function f(t), f(t)𝛿(t) = f(0)𝛿(t)
Because for everywhere but 0, 𝛿(t) = 0
Integrating:
∫f(0)𝛿(t) dt = f(0) ∫𝛿(t)dt = f(0)
Because f(0) is a constant and ∫𝛿(t)dt = 1
This winds up being the same as “sampling” the function at t=0.
More generally, to sample at an arbitrary time T, just shift the Impulse function:
∫ f(t) 𝛿(t-T)dt = f(T)
Useful Signal Operations:
List of Basic Operations
- Time Shifting
- Delay
- Advance
- Time Scaling
- Compress
- Expand
- Time Reversal/Reflection
- Shift
- Scaling
- Sum
- Product
Useful Signal Operations:
Time Shifting Operations
Delay signal by T:
xd = x(t + T)
Advance signal by T:
xa = x(t - T)
Useful Signal Operations:
Time Scaling Operations
Compress signal by a factor of T:
xc = x(t * T)
Expand signal by a factor of T:
xe = x(t / T)
Useful Signal Operations:
Time Reversal
Reverse the signal, reflecting x(t) across the horizontal axis:
xr(t) = x(-t)
Signal Operations:
Split this function into Even and Odd components:
x(t) = e-atu(t)
The Even component is the function divided in half, and one half made the reflection:
xe(t) = (1/2)[e-atu(t) + eatu(-t)]
The Odd component is the function divided in half, with one half made a reflection with sign change
xe(t) = (1/2)[e-atu(t) - eatu(-t)]
