Frequency and Bandwidth
A means of quantifying and interpreting
media capacity
Sinusoidal Waves
• Taught in math as a function of an angle– sin(90o) =1
• Taught in Physics as a function of position in wave propagation
• Think of a wave propagating, – sinusoidal if consider f(t), one point over time– sinusoidal if consider f(x), snapshot for all x
• Our concern is f(t) with x fixed at each end
Physics Wave Machine
)sin(t
2
-1
1
t
Function goes through ONE cycle in
2 pi seconds.
)2sin( t
-1
1
t.5 1.0
Function goes through ONE cycle in ONE second.
FREQUENCY = 1
Why use ft2
Let f=1. )2sin( ft Goes through 1 cycle as t=0..1.
t ft2 )2sin( ft
0 0 0
.25 2 1
.5 0
.75 23 1
1.0 2 0
?
fAs t=0..1, =0.. cyclesft2
1 2 1
2 4 2
3 6 3
10 20 10
=
)8sin( t
has
f=4
Answerby
inspection
Three basic features of a wave
)2sin( ftA
Amplitude phasefrequency
The only variable (t):As t -> 0..1 the functiongoes through f cycles
Amplitude
Frequency
Phase
Focus on A and f
• Phase represents a shift right or left in the signal.
• This is a timing issue.
• Sin and cos only differ in phase (the time at which you examine the wave
• Our focus is on A and f
You should know
• The impact of changing the amplitude, A, of a signal.
• The impact of changing the frequency, f, of a signal.
• The impact of changing the phase of a signal.
• How to calculate the frequency of a sin wave.
Additive and SubtractionProperties
• Signals can be expressed as a sum of sinusoidal signals
• One can subtract frequencies and effectively filter the signal
• How to build filters is not important to this course, but the concept of filtering is.
• (see other graphs)
Frequency 1
sin(2*Pi*t)
Frequency 3
1/3sin(6*Pi*t)
Frequency 5
1/5sin(10*Pi*t)
Sum of First 3 termssin(2*Pi*t)+1/3sin(6*Pi*t)+1/5sin(10*Pi*t)
Fourier Series• Surprisingly all periodic signals can be
expressed as a sum of sinusoidal signals
• See examples of sawtooth, rectified cos, etc.
• MOST IMPORTANT TO US:– A square wave (fundamental of a digital signal)
can be expressed as a sum of sins.– Requires INFINITE number of terms to exactly
express a square wave– see example and program for seeing tradeoff of
sin terms versus squareness
Frequency vs Time domain
)10sin(5
1)6sin(
3
1)2sin( ttt
S(f)
1
f1 2 3 4 5
.33.2
)10sin(5
1)6sin(
3
1)2sin( ttt
S(f)
1
f1 2 3 4 5
.33.2
Redundant except for phase information in the time domain
Stereo AmplifierApplication of frequency analysis
Amplifier(lower frequencies)
INPUT OUTPUT
Certain frequenciesdo not pass through
Frequencies within the dashed box are uniformly amplifiedThis defines a transfer function for the amplifier: S(f)
Communications media have similar characteristics and distortcertain frequencies like the amplifier.
Media is the same
Ethernet Cable
INPUT OUTPUT
Certain frequenciesdo not pass through
What happens when you limit frequencies?
Square waves (digital values) lose their edges -> Harder to read correctly.
Graphing Application
• Vary the number of terms and regraph the series.
• Increasing number of terms-> more square
• Decreasing number of terms-> less square
• Decreasing terms is analogous to passing the signal through a filter and has the effect of distorting the signal.
Frequency Windows Application
The fundamental problem
• Undistorted signal
• Noise, bandwidth limitation, delay distortion, etc changes signal
• Receiver must determine when to read
• Receiver must correctly read
• Increasing noise-> Increased probability of misreading
• See overhead
Encoding
• Lots of techniques for encoding information
• Based on – type of data
• digital
• analog
– type of medium• digital
• analog
Way of looking at techniques
Data
MediumDigital
Analog
Digital
Analog
NRZManchesterDifferential Manchester
Phase Coded Modulation(digitized voice)
ASKFSKPSK
modems AM/FM radioTelevision