Wireless Data Technologies
Introduction
Omega - Angular
Frequency
Amplitude Modulation
AM Applet
Frequency Modulation
FM Applet
Conclusion
Omega - Angular Frequency
Angular frequency is the rate of change of an angle per unit
time. It is often denoted by the Greek symbol omega, w .
To calculate angular frequency it is necessary to multiply the
frequency of a signal, f by 2 pi.
Pi is roughly equal to 3.14159. It is the ratio of the circumference of
a circle to its diameter.
w = 2pf
Angular frequency is measured in radians per second, with
dimensions T −1
To carry out calculations with regard to
modulation, it is necessary to use angular frequency.
Amplitude Modulation
AM is the simplest method of modulation. A carrier wave of suitable
frequency is chosen to suit the application required.
All radio stations in the Long and Medium wave band use AM. The music
quality is fairly poor as the bandwidth allocated for each station is
only 9kHz. AM is also susceptible to picking up external noise from
power equipment and atmospheric conditions such as lightning.
This was the first type of modulation used for communicating
signals
from one point to another and is the simplest to understand.
The signal can be written as:-
y(t) = (C + M sin (wmt
+ f ) sin wct
)
This represents a signal at frequency
wc/2p
Hz
whose amplitude is modulated by another frequency
wm/2p
Hz.
m = am/ac is the modulation index.
To find the frequency spectrum of the AM signal the
above expression can be rewritten as below:-
y(t) = ac{cos wct
+ m/2(cos(wc
+ wm)t
+ cos(wc
- wm)t)}
We can see from the above expression that the frequency spectrum
consists of
3 components at frequencies:
- wc/2p Hz
- (wc
+ wm)/2p
Hz
- (wc
- wm)/2p
Hz
Above can be seen the frequency spectrum of an AM signal. Note that the
carrier wave still exists at the centre of the two sidebands. The
sidebands will move towards and away from the carrier as the signal to
be modulated varies. The higher the signal's frequency, the further the
sidebands will be from the carrier, the lower the signal frequency, the
closer the sidebands will be to the carrier wave.
This cannot be seen from the signal when it is viewed in the time
domain.
The applet below shows an amplitude modulated signal in blue and
how it depends on m and wm.
The modulating signal is shown in red and the frequency
spectrum in black.
Try adjusting the sliders to see how the output modulated signal varies
when input parameters are changed.
AM
Applet
In the applet below, you can see the effect of changing the modulation
index, the modulating signal and the carrier frequency.
Frequency Modulation
Frequency modulation is used for short distance, high quality
radio signals. Interference is rarely heard when using FM as the effect
of noise is to produce a spike on the signal which raises the signal's
amplitude. However, the receiver is trying to look at the variation in
the frequency of the arriving signal as it is these variations that
carry the information that we wish to transmit.
Today, FM is used to carry high quality music and speech. FM is line of
sight and therefore does not travel far. The FM spectrum for radio
transmission is around 100 MHz.
In frequency modulation the transmitted signal's amplitude is kept
constant and the
frequency is modulated by the instantaneous amplitude of the modulating
signal e.g. music or voice.
The modulation index, m, for FM is:
m = maximum frequency
deviation/modulating frequency.
An FM signal can be represented
as:-
y(t) = ac sin(wct
+ m sin wmt )
The frequency spectrum can be found by rewriting the above
expression as a sum of components of constant frequency using
the properties of the Bessel Functions. This gives:-
y(t) = ac{Jo(m)
sin(wct)
+ J1(m)[sin(wc
+ wm)t - sin(
wc
- wm)t]
+ J2(m)[sin(wc
+ 2wm)t + sin(
wc
- 2wm)t]
+ J3(m)[sin(wc
+ 3wm)t - sin(
wc
- 3wm)t]
+ ...
From this expression it can be seen that the FM spectrum consists of a
component at a frequency of wc/2p
Hz and an infinite number
of lines at (wc
± nwm
)/2p
Hz and that the amplitude of the components are given by
the Bessel
functions. The applet below shows the modulating signal in red, the FM
signal in blue,
the frequency spectrum in black and how they vary with the modulation
index, carrier frequency and modulating frequency.
Move the sliders to discover the way the FM modulated signal changes as
the input parameters change.
FM Applet
The applet helps to display the characteristics of FM and allows
us to see how the waveform and the spectrum change when the variables
are adjusted.
Conclusion
Angular frequency is used when we want to measure the the
rate of change of an angle per unit
time. It is measured in Hertz.
AM is produced by multiplying the input signal by the carrier wave.
There are TWO distinct sidebands produced when using AM. AM is used for
low quality broadcasts and is susceptible to induced noise.
FM is produced by adding the frequency of signal we wish to carry to
the frequency of the carrier wave and taking the sine of the result.
FM may have many sidebands and is used for high quality broadcasts as
it does not suffer badly from noise.