6. The Time Dimension I: Sampling
6.1 Frequency Spectrum
All signals which are continuous functions of time have a frequency
spectrum. This is essentially a representation of the signal in terms of
its rate of change with time, i.e. the speed with which its time profile
or morphology changes.
Any periodic function of time, i.e. one which has a repetitive or cyclical
nature consists of frequency components that are integral multiples of
the rate of repetition of the signal. This is known as the frequency
spectrum of the signal. An example of this is shown below in Fig. 6.1.
V
V
↔
….
0 fR 2fR 3fR 4fR 5fR 6fR
t
f
T = 1/fR
Fig. 6.1 The Frequency Spectrum of a Periodic Signal
Fourier Series
The theory behind Fourier’s Series states that any periodic signal can
be represented as a summation of sinusoidal components at the
fundamental frequency of repetition of the signal and harmonics of
this frequency.
The magnitudes of the individual components can be calculated
mathematically if a mathematical description of the signal in time is
available. Consider the bipolar square wave shown in Fig 6.2 below.
This is a symmetrical periodic waveform which has an amplitude of +A
Volts for the first half of the cycle time from t = 0 until t = T/2 and an
amplitude of –A Volts for the second half of the cycle from t = T/2 until
t = T.
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The waveform is described
as a function of time as:
f(t)
A
f(t) = { A, 0 < t ≤ T/2
{ -A, T/2 < t ≤ T
T/2
0
T
0
-A
T
Fig. 6.2 The Time Profile of a Bipolar Square Wave
The Fouier Series is given as:
f(t) 
4
1
1
A[Sin2 ft  Sin2 3ft  Sin2 5ft  ............]

3
5
Fig. 6.3 Partial Fourier Components of a Bipolar Square Wave
2
t
The Fourier series of a square wave only has odd harmonic
components. Fig. 6.3 shows the first three Fourier components of the
square wave as well as their summation. The summation bears a
reasonably close resemblance to the original square wave but it can be
seen that there is some oscillatory variation around the pulse
amplitude due to the finite number of components summed. If more
components are added, the waveform becomes increasingly more
close to the ideal square wave as can be seen from Fig. 6.4.
Fig. 6.4 Varying Numbers of Harmonic Components in the Fourier
Series
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Usually the components at higher frequencies have lower magnitudes
than those at lower frequencies so that the amount of energy in the
spectrum decreases with increasing frequency. This means that in
general the higher frequency components contribute much less to the
overall signal profile than the lower frequency components. However,
they do influence small local changes taking place in a short time span.
When a signal is a continuous function of time but is not a periodic or
recurrent function, it still has a frequency spectrum. However, the
frequency spectrum is not composed of defined harmonic frequencies
but rather has all frequencies present, usually up to some maximum
frequency of interest, fM, as seen in Fig. 6.5. The magnitudes of
components at different frequencies are continuously varying, as are
any harmonic relationships present. Consequently, the spectrum is
simply shown as a continuous shaded spectrum, up to the maximum
frequency of interest. This is referred to as the Baseband Spectrum.
The shape of the spectrum shown in Fig. 6.5 is arbitrary as this varies
with time and is only for illustrative purposes.
V
V
↔
t
0
fM
f
Fig. 6.5 The Frequency Spectrum of a Non-Periodic Signal
Bandlimiting:
In practice, the higher frequency components in a spectrum at low
amplitudes tend to get contaminated by noise and do not really
contribute much information to the signal. Consequently, they can be
omitted without loss of information while at the same time reducing
the amount of noise present in the signal.
In Analogue-to–digital conversion the highest frequency present is
deliberately limited to a maximum, a process known as bandlimiting.
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This is done so that the highest frequency present can be guaranteed
not to exceed a specified maximum limit. Bandlimiting is accomplished
in practice by passing the baseband signal through a low-pass filter
which ideally passes frequency components below its cut-off
frequency, set to fM, and suppresses all components above this
frequency.
6.2 The Principle of Sampling
Introduction:
The discussion up to this point has centred on the particular value of a
signal, its quantisation and encoding into binary form, and the issues
of accuracy and resolution surrounding this. Changes in the signal
voltage have only been considered in absolute terms and not as
changes with time. In the real world all signals, and in particular
electrical ones are functions of time, i.e. they are time varying as
shown in Fig. 6.6.
Moreover, in the real world every process takes time to achieve and
this applies to data conversion also. This means that the quantisation
of any signal value and its encoding into binary form takes a finite
amount of time to accomplish. It cannot be done instantaneously. This
means that, just as we cannot quantise the signal amplitude with
infinite resolution of its magnitude, we cannot convert the signal with
infinite resolution in time either. It can only be converted by taking
samples of the signal, usually at regular intervals in time, i.e. at a fixed
finite rate.
Relative
Amplitude
Time
Fig 6.6
A Continuous Signal as a Function of Time
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Sampling:
This leads to the concept of sampling, where the absolute value of the
signal voltage is sampled at regular intervals in time, nTS , as shown in
Fig. 6.7. The time in between samples is then used to encode the
samples into binary form and store or transmit them. The sampling
process results essentially in series of sample values which are
updated at a regular rate as shown in Fig. 6.8.
Voltage
0
TS
2TS
3TS
4TS
5TS
6TS
7TS
8TS
Time
Fig 6.7 Sampling of a Continuous Time Signal
Voltage
0
TS
2TS
3TS
4TS
5TS
6TS
7TS
Fig. 6.8 The Result of the Sampling Process
6
8TS
Time
The Sampling Theorem:
The Sampling Theorem, originally proposed by Nyquist, states that: ‘All
of the information present in a time varying signal is contained within
samples of the signal taken at regular intervals in time at a rate which
is greater than or equal to twice the highest frequency component
contained within the spectrum of the signal.’
That is, if a signal which is a function of time, f(t), is bandlimited to
contain a maximum frequency component of fM , then all of the
information present in the signal can be recovered from samples of the
signal taken at a frequency fS , where:
fS  2fM
The frequency fS is known as the Sampling Frequency and a value of
this frequency of fS = 2fM is known as the Nyquist Sampling Rate.
It can be seen in Fig. 6.9 below that if there are at least two samples
per cycle of the highest frequency component present in the signal,
this is sufficient to characterise and later recover this component. If
the sampling frequency is exactly equal to twice the highest frequency
present then there is a danger that the samples could be synchronised
with the zero crossing points of this component and would give a
sample value of zero. In order to prevent this happening, the sampling
rate is normally made a little higher than the theoretical Nyquist rate,
so that the sampling process is not synchronised with any of the
frequency components present in the signal.
V
sinewave at highest
frequency present in
the waveform
sampled at Nyquist
rate
V
0
recovered
sinewave at
highest
frequency
present
0
t
t
Fig. 6.9 Sampling the Highest Frequency Component at the Nyquist
Rate
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Sampled Signal Spectrum:
When a signal is sampled at a rate or frequency of fS ≥ 2fM, then the
frequency spectrum of the sampled signal contains the original
baseband signal spectrum and images of this spectrum symmetrically
located about harmonics of the sampling frequency as shown in Fig.
6.10.
V
V
baseband
spectrum
image
spectrum
↔
t
TS = 1/fS
Fig. 6.10
0
fM fS-fM
fS
fS+fM
2fS-fM
2fS
2fS+fM f
Frequency Spectrum of a Sampled Signal
Aliasing:
If the signal is sampled at a rate which is less than the Nyquist rate,
i.e. fS < 2fM, which means that fS - fM < fM, then this results in an
overlap of the image spectra as shown in Fig. 6.11 and Fig. 6.12 below.
This results in distortion of the recovered signal known as Aliasing
Distortion so that the time profile of the recovered signal is different
from the original input signal which was digitised.
V
V
↔
t
TS = 1/fS
Fig. 6.11
0 fS-fM fM
fS 2fS-fM fS+fM 2fS
2fS+fM …. f
Aliasing Distortion Due to Overlap of Spectra
aliasing
distortion
V
Fig. 6.12
Overlap of Baseband and Image Spectra
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T
i
m
e
6.3 Aliasing Distortion Speech:
The following examples are samples of speech all quantised with 8 bits
resolution but varying sampling frequencies relative to the Nyquist
rate where the phrase spoken is:
‘The possibility of a Mann Act conviction, resulting in disbarment
proceedings and total loss of his livelihood, was a key factor in his
decision‘.
Speech sampled at Nyquist Rate: Speech-Mann Act 22.05kHz.wav
Speech sampled at 1/2 Nyquist Rate: Speech-Mann Act 11.025kHz.wav
Speech sampled at 1/4 Nyquist Rate: Speech-Mann Act 5.5125kHz.wav
Speech sampled at 1/8 Nyquist Rate: Speech-Mann Act 2.75kHz.wav
The following examples are samples of speech where the phrase
spoken is:
‘Rimmer I’m bored…... Bored?, This is essential routine maintenance.
It’s absolutely vital for the well-being of this crew, this mission and
this ship……. Dispenser 172 – Chicken Soup nozzle clogged!’.
Speech sampled at Nyquist Rate: Speech-Red Dwarf 22.05kHz.wav
Speech sampled 1/2 Nyquist Rate: Speech-Red Dwarf 11.025kHz.wav
Speech sampled 1/4 Nyquist Rate: Speech-Red Dwarf 5.5125kHz.wav
Speech sampled at 1/8 Nyquist Rate: Speech-Red Dwarf 2.75kHz.wav
As can be heard the speech becomes more distorted the more the
Nyquist sampling rate is infringed and the greater the amount of
aliasing present. However, the nature of the distortion is different than
that produced by low quantisation resolution. It affects intelligibility
much more directly because it alters the relative magnitudes of
frequency components. This begins at the higher end of the frequency
spectrum but works downwards as the sampling frequency is lowered.
However, there is more to intelligibility than simply quantisation
resolution. See:
http://www.youtube.com/watch?v=-V2MfBoBiyw
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Music:
Note that in the following examples the Nyquist rate is higher than in
the speech samples because of the wider baseband spectrum.
Music at Nyquist Rate: Music-Eleanor Rigby 44.1kHz.wav
Music at 1/2 Nyquist Rate: Music-Eleanor Rigby 22.05kHz.wav
Music at 1/4 Nyquist Rate: Music-Eleanor Rigby 11.025kHz.wav
Music at 1/8 Nyquist Rate: Music-Eleanor Rigby 5.5125kHz.wav
Music at 1/16 Nyquist Rate: Music-Eleanor Rigby 2.75kHz.wav
As with quantisation resolution, music signals are much more sensitive
to the effects of aliasing. This is again because there is much more
energy in a music signal at higher frequencies than is the case in a
speech signal. The higher frequency energy defines subtle changes in
the signal which affect the harmonic quality of what is heard and is
appealing to the listener in a musical context. Therefore a lesser
infringement of the Nyquist sampling rate has a more perceptible and
disagreeable effect than in the case of speech.
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