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