Contents
Sampling
Introduction
Nyquist Sampling
Natural Sampling
Flat topped Sampling
Aperture Error
Aliasing Revisited
Quantization Distortion
Conclusions
Sampling
Introduction
A great deal of signal processing is required to convert an analog signal to the digital
domain. This collection of SystemView models will examine this process, and its
consequences. Some of these include:

Natural & flat topped sampling

Aperture error

Aliasing

Quantization distortion
Nyquist Sampling
In any sampling process, it is necessary to obtain enough samples to be representative of
the whole. The Nyquist theorem states that the minimum number is 2 samples per Hertz.
In actual practice it is often necessary to take more than this minimum amount.
Sampling
In a telephony codec, the ratio is
8 k samples
 2.35 samples / Hz .
3.4 kHz
Aliasing occurs if less than 2 samples/Hz are obtained. Notice what happens in the time
domain, if a sine wave is sampled at less than the Nyquist rate:
Sampled Signal
Aliasing Signal
Sampling P oints
Reminder
Aliasing also occurs in computer simulations if the System Clock is not set to a high
enough rate for the model. In most cases, choose a value of about 5 – 10 times the highest
frequency in the model.
There are two types of sampling: natural and flat-topped. Of these two, natural sampling
is the easier to analyze. However, in practice, flat-topped sampling is used in an ADC.
Natural Sampling
In this type of sampling, the resultant signal follows the natural shape of the input during
the sampling interval.
Switch
Analog input signal
Sampled output signal
The sampling function can be regarded as a form of multiplication. An output occurs
when the input is multiplied by 1, but nothing emerges when it is multiplied by zero.
x



=
T
Sampling
The pulse signal acts as a sample gate for the analog waveform. This process, known as
PAM† is actually time domain multiplication.
A 1 Hz sinewave sampled at 10 samples per second, with an aperture of 20 mSec
resembles:
sinc envelope
Baseband Spectrum
Amplitude
The spectrum of a sampled baseband creates a whole host of sampling artifacts.
f
Sampling Signal
Equal amplitude sidebands
1/ 
m
fs - f
m
f
2f
4f
s
s
2/ 
3f
s
s
5f
s
3/ 
4/ 
6f
s
5/ 
Frequency
fs + f
m
Flat topped Sampling
The sampled signal is held constant in a DAC during the conversion process. This alters
the time and frequency domain components.

X
Input Signal
The simulation results are:
†
Pulse Amplitude Modulation


=
T
Sampling Signal
Sampled & Held Signal
Sampling
The spectrum of this waveform resembles:
Notice that the results are almost the same as with natural sampling except that the
frequency pairs are not equal in amplitude. Instead, each component follows the sinc
envelope.
This subtle difference is due to aperture error. It’s affect on a sampled signal becomes
more apparent when a baseband is sampled.
Sinc Envelope
Baseband Spectrum
Amplitude
The spectrum after sampling a baseband resembles:
f
Sampling Signal
Sidebands follow the sinc envelope
1/ 
m
fs - f
m
f
2f
4f
s
2/ 
s
3f
s
s
3/ 
5f
s
4/ 
6f
s
5/ 
Frequency
fs + f
m
The spectrum of a flat-topped sampled waveform follows the sinc envelope. The
amplitude of the highest frequency components have been adversely affected. This
constitutes aperture error in the frequency domain.
This phenomenon can be significantly reduced by using non-return to zero sampling.
NRZ
Sampling
Sampling a Baseband
Notice the effect of flat topped and natural sampling on a baseband signal:
The high frequency content of a baseband signal is affected by flat-topped sampling since
its spectrum follows the sinc envelope. This distortion can be reduced by either increasing
the sampling frequency or reducing the sampling aperture width.
Aperture Error
Aperture error is the difference between the actual value of the input signal, and the flattopped sample value.
The magnitude of this difference is related to the input frequency and sampling width.
Aperture Error
Input Waveform
Aperture Error
Sampling
Waveform

Time Domain
Baseband
Spectrum
Sinc Envelope
Time
Frequency
Frequency Domain
The error caused by a 20 mSec aperture sampling a 1 Hz signal 10 times a second
resembles:
Sampling
This error or noise can be reduced by reducing the aperture width.
Aperture Error
Input Waveform
Aperture Error
Sinc Envelope
Sampling
Waveform

Baseband
Spectrum
Time
Frequency
Time Domain
Frequency Domain
The error caused by a 10 mSec aperture sampling a 1 Hz signal 10 times a second
resembles:
Notice that the aperture error was reduced when the sampling window was reduced.
It would appear that reducing the sampling aperture to near zero would eliminate this
form of distortion. This would be true except for the fact that at the other end of the
digital transmission system, the signal must be reconverted back to digital. In the case of
a telephony codec, the sample rate of 8 KHz produces an effective output aperture of 125
Sec at the DAC.
Video codecs can reduce aperture error at the DAC by interpolating between samples.
Aliasing Revisited
Another way to examine aliasing is when the lower sideband associated with the
sampling frequency, overlaps the baseband spectrum.
Baseband Spectrum
LSB
fs
2
Aliasing Noise
USB
fs
Sampling Frequency
Besides creating tones, foldover distortion can create broadband aliasing noise.
Aliasing can be reduced by increasing the sampling rate, and/or by limiting the input
baseband frequency prior to sampling by means of an anti-aliasing filter.
Sampling
Quantization Distortion
Noise associated with step size is known as quantization noise.
1100
110
1011
1010
101
Reduced
Quantizing
Noise
Digital Code
Digital Code
1001
100
011
1000
111
110
101
100
010
011
001
010
001
000
000
0
1
2
3
4
Sampling Instant
5
6
0
1
2
3
4
5
6
Sampling Instant
It is relatively easy to determine the amount of quantizing noise present if linear signals
are digitized. The amount of noise is equal to the difference between the input analog
signal and the quantized signal.
This effect can be reduced by increasing the number of bits per sample, thus reducing the
quantization step size.
In telephony codecs, the step size is adjusted according to the size of the audio signal.
This process, called companding, helps to keep the signal to quantization noise ratio more
or less constant. Small signals effectively have a 12-bit resolution whereas large signals
have 8-bit resolution. This takes advantage of the psychoacoustic characteristics of human
hearing.
Sampling
Conclusions
This model was able to demonstrate the affects of aperture error on a single frequency
input in both the time and frequency domains. In order to demonstrate this phenomenon
on a baseband signal, a more comprehensive model, perhaps using sweep tokens at the
input would be necessary. Never the less, this model demonstrated that:

Aperture error occurs whenever flat–topped sampling is used

The magnitude of the error is directly related to the aperture width

Aperture error adversely affects the high frequency components in the sampled
signal