Transcript Oversampling - Dr.Mohammed Saleh Al.Ansari

Example of Oversampling
Aims and learning outcomes
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To demonstrate how oversampling can improve the quality of
conversion for acoustic signals.
Students will be able to:
contrast the representations: PCM and DPCM;
describe the components of an oversampling ADC;
describe the advantages of using differential coding and
oversampling convertors;
describe noise shaping as a system to reduce S/N ratio;
calculate the improvement in S/N ratio for a simple sigma
DPCM convertor
Introduction
Oversampling is sampling at a higher rate than is strictly
needed. It is very popular in quality digital audio equipment.
Advantages include less circuitry (cheaper) and better S/N
ratio. It allows less expensive circuits to be used, as the
sampling frequency is traded off against the number of bits
needed to represent the signal. Fundamental to oversampled
convertors, is the representation of signals in Differential
Post Code Modulation (DPCM), so this is where we will
start.
What is DPCM?
Definitions
PCM is the familiar digital representation of an analog signal. It
consists of a series of pulses, evenly spaced in time, the pulse
heights being proportional to the analog signal at the sample
time.
DPCM is formed from the difference between adjacent PCM
signal values
If the PCM signal is PCM x1, x2, x3, .... xn, then the DPCM
signal is (x2-x1), (x3-x2) .... (xn - xn-1)
DPCM is effectively numerical differentiation of a PCM signal,
hence the name.
Exercise for students
1)
Three samples of a PCM signal are 0.15, 0.25, 0.57, 0.1, -0.1
volts, the first signal is at t=0 and the time interval is 1x104s. Calculate the
equivalent DPCM from t=0
to t=4x10-4s.
2) Sketch the DPCM
equivalent of the PCM
signals shown in two
figures to the right. (assume
the sampling frequency
is 10 kHz.)
Answers
1)
time (ms)
0
DPCM (volts) Undefined
0.1
0.1
0.2
0.32
0.3
0.4
-0.47 -0.2
2) The middle figure gives a signal like (pk=pk amp. 4V).
The right figure gives a DPCM signal of Acos(2Bft) where A is
0.6283 volts.
Why and more on DPCM
Why DPCM ? Because it gives a better S/N ratio in
convertors by using a process of noise shaping – see below.
As we are now converting the difference between adjacent
samples, rather than the samples themselves, there is no
longer a limit to how large a signal the system can represent.
The limiting factor is, however, how big a difference
between adjacent samples it can represent - so system is
limited by its slew rate. To encode large differences between
adjacent samples, the convertor can either increase the
number of bits that the quantizer produces, or increase the
sampling frequency.
Analog in
Digital out
Not slewing fast enough
Double number of bits
Double sample rate
The signal output from a DPCM falls at 6dB/octave compared
to a PCM signal. (it is limited by the slew rate). The noise
floor due to quantization error and dither is constant across all
frequencies.
Obviously, this signal roll off (shown right above) is
undesirable, so the signal is integrated (shown below right).
This tilts the noise floor, so noise is greater at high frequencies
- hence the sigma DPCM converters are known as noise
shaping converters. This will be exploited in the oversampling
systems we will discuss.
Oversampling
Figure 1 illustrates that in a digital system we can play off
the number of bits against the sampling frequency. If we
lower the number of bits representing a signal, we can
still get an accurate rendition of the input signal provided
we increase the sampling frequency (the number of
steps). This fact is key to the use of oversampling
converters which are commonly used on quality audio
systems..
Level
6dB/octave
Signal
Noise
F (Hz)
Level
Signal
Noise 6dB/Oct
F (Hz)
Consider a 4x oversampling system for an audio system designed
to work to 20kHz.
Non-oversampling system:
Analogue signal
 anti-aliasing filter 24dB/oct, -3dB point 18kHz
 AD convertor
 Digital out, fs = 40kHz, 16 bit
Oversampling system:
Analogue signal
 anti-aliasing filter 6dB/oct
 AD convertor (14 bit, fs=160kHz)
 Digital low pass filter, 24dB/oct, -3dB point 18kHz
 Digital filter to decrease sample rate
 Digital out, fs = 40kHz, 16 bit
Analog in
Anti-aliasing
1 bit convertor
+encoder
Nx44.1 kHz
LPF digital Sample rate
16 bit PCM
filter
reduction by N
44.1 kHz
Oversampling is used because:
• Analogue anti-aliasing filters are expensive due to the high
quality analogue components needed. In an oversampling
system, the analogue anti-aliasing filter can have a very slow
roll off and be very cheap.
• When oversampled, the noise floor rises with frequency (see
figure above). Consequently, this noise can be filtered out by
the digital low pass filter as it is outside the audio frequency
range. In this 4x oversampling system the signal to noise ratio
is improved by 12dB. This is a process called noise shaping.
We can use less circuitry for the convertor and so reduce the
cost. As the sampling frequency increases the number of bits
we need to hold the same information decreases. By increasing
the sampling frequency, we can reduce the word length used in
conversion.
Disadvantage of oversampling
• The data storage is much more inefficient,
but with SACD and similar technology
oversampled data is now being stored
directly (i) because computer storage
now cheap (ii) removes up and down sampling
stages which are detrimental to quality.
As we can trade off sample rate verses number of quantization
bits in a DPCM system - see above - there is no reason why we
can’t go to the extreme of only having a one bit
quantizer.
Input
X
+
U
Integrator
V
1 bit quantizer
Y
Out
This is delta modulation.
Z
The output is simply +1 or 0
depending on whether the previous signal values are less than
or greater than the current value.
-1
It obviously has to work at an extremely
high sample rate. One advantage is that the
conversion circuit can be very simple.
This produces single bit DPCM information. If we then
integrate the output from the delta modulator, we get PCM.
Such a convertor is known as a delta-sigma modulator.
(shown above)
So when considering the effects of sampling in an audio
system, the simple considerations of quantisation error, and
the simple formulations for dynamic range given before are
not correct. Previously, we have assumed that any
quantisation error produced stays within the audio band, but
in an oversampling system the error is deliberately shifted out
of the audio range so that it can be removed by low-pass
filtering in the digital domain. If 16 bit convertor can be
replaced by a delta-modulator, than the noise in the audio
range decreases by a factor of (216 ≡ 48dB!) compared with
non-oversampling.
MATLAB tasks
Quite a few concepts have been introduced so far. Now
it is time to show how realistic these are by doing some
MATLAB simulations. (Incidentally, real researchers
developing convertors use these sought of MATLAB
demonstrations).
In the course notes, the one-bit convertor was shown
(figure above). It may not be immediately obvious that this
achieves noise-shaping and so improves signal to noise ratio.
The intention of this tutorial exercise is to prove its
operation through the MATLAB script oversampling.m. Run
and examine the workings at output of the script.
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There are many things to note, but of primary interest:
Note that a 1 bit signal does really convey frequency
information (Figures 2 and 3). This can seem fantastical, but
you can see it in the time picture (if you stare at this long
enough you can also see a picture by Bridget Riley).
Figure 3 shows the input frequency and all the additional
frequencies generated by quantisation. Notice the rising
noise floor due to this being a noise shaping convertor. The
frequencies of interest are to the far left of the graph, and so
the noise >16kHz is going to be lost when we downsample.
Figure 4 shows the input and output signals. Notice they are
not identical. The sine wave is distorted by the quantisation
– some of this is the effect of dither, the rest is the rounding
errors.
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Figure 5 shows the final spectrum of the converted signal –
the noise floor is too high to be useful, higher order
integrators are needed.
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Listen to the sound. At the end of the script the signal before
and after conversion is played. (These are played
alternatively 5 times). Notice that you can hear the dither
noise on the after conversion signal, but there is no tonal
change in the pure tone. Remove the dither from the
convertor by commenting out the relevant line. Rerun and
listen. Notice that the tonal changes, this is the problem of
signal dependent distortion as was found for a nonoversampled case. Look for evidence of distortion in the
graphs.
Oversampling in DAC
16 bit
44.1 kHz
Digital
Interpolating
digital filter
4x oversampled
28
28

re-quantizer
DAC
Reconstruction filter
14
Z-1
14 MSB
14 LSB
Analog out
Figure 1 Noise shaping in DAC - after Watkinson pg157, Z-1
means delay one sample
The process of error interaction leading to noise shaping is also
used in DA conversion.
The interpolating filter generates 28 bit word length, which is
then requantized to 28 bits before being transmitted through a
DAC, reconstruction filter and then is the analog output signal.
The 14 LSB from the requantizer are delayed by one sample and
added to the interpolating digital filter output.
The 14 LSBs from the last sample (the quantization error) and
28 bits fr om the present sample interfere. Consequently we no
longer have
a white noise spectrum but something different.
The RH graph was generated aong these Noise
principles using a spreadsheet. The
16 bit
bottom line is a triangular dither
quantization error spectrum – which
18 bit or 16 bit
is white noise as expected. The top line with oversampling
has been noise shaped by adding
Audible range
Frequency
the quantization error of the
previous sample to the signal before quantization.
A rising noise floor can be seen. If this
rising noise floor shifts the noise beyond the
audible range – which is possible in an
oversampling system. This quantization error
noise is then lost and inaudible. Better S/N ratio.