in this case, no aliasing occurs and the decimation ...sdblunt/644/coursenotes5c.pdf · note that...
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5-32
In this case, no aliasing occurs and the decimation equation can be expressed as:
Since aliasing is very undesirable, the decimator is often preceded by a lowpass filter which insures
that aliasing will not occur
for 1
MXM
Y
Mh(n) y(n)x(n)
w(n)
where
otherwise 0,M
,1
H
H
M
M
ideal LPF
5-33
)(CX )(CX
)()( TXX CS
)(X
)(X
)(dX
)(dX
)( TX d
)(dH
)()()(~
XHX d
)(~
dX
5-34
Increasing the Sampling Rate by an Integer Factor
When the sampling interval is decreased by an integer factor, L
then the sampling rate is increased by the same factor
Now suppose we have a sequence x(n) which has been obtained by sampling x(t) at t = nT.
And we want a new sequence y(n) which could have been obtained by sampling x(t) at t = nT '. This
implies that we must interpolate L–1 new sample values between each pair of sample values of x(n).
T'T
L
Fs'
1
T 'L
T LFs
nTt
nnxnTxtx
=
<<-for
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Therefore, TxTnxnTyLn
integer)an is (i.e., ,20for Ln
Ln LL, , n=xny
and y n 0 otherwise
Lx(n)
y(n) x nL
A finite sequence
L = 2 (i.e., zero-fill between
each sample of x(n))
A new sequence, but the up-
sampler does not give us this. So
where does it come from?
x(n)
Lnxny
)(nw
n
n
n
5-36
To answer this question let's examine the interpolation process in the frequency domain.
n
njenyY
n
nj
Ln exY
LXY
L
nxny where
so
r
LrjerxY then
therefore
Note that while decimation implied a shifting and scaling of the frequency axis, up-sampling implies
only a scaling of the frequency axis.
To complete the interpolation process, some mechanism must be included to filter out the unwanted
images so that the output signal sequence truly looks as if it were sampled at a higher rate.
integeran is whereL
nr
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The complete system is
L h(n)
y(n)
x(n) w(n)
LPF
otherwise 0,L
,
LH
An illustration of the process follows:
x(n)
)(ny
)(X
)(Y2let L
rateNyquisttheatsampledoriginally)(tx
n
n
6 4 2 2 4 60
T
1
T
1
3 2 2 30
2
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Low-pass filter)(H
)(W
)(nw
n
L
'
1
TT
L
22
2 30
2
22
2 30
2
4
Low-pass filtering removes the unwanted images.
5-39
An interesting example is linear
interpolation. It is not highly accurate but it
is simple and popular.
)(CX
)(X
)()( LXX e
)(H
)()()(~
eXHX
)(H
)(linH
5-40
Changing the Sampling Rate by a Non-Integer Factor
L MLPF
Gain = 1
Cutoff = /M
LPF
Gain = L
Cutoff = /L
ML
L
T
L
T
L
T
L
MTT
LPF
Gain = L
Cutoff =
min{/L, /M}
TL
T
L
T
L
MT
Sampling period
Sampling period
nx
nx
ny
ny
For example, if we want T’ = 1.01T, then let L = 100 and M = 101
These two linear filters can be implemented as one
5-41
Here’s an example from the text:
Change the original sampling rate by a non-integer
factor:
32 nx ny nh
H
3
3
2
)(CX
)(X
)(eX
)(dH
)()()(~
ed XHX
)(~
dX
5-42
5.6 Multirate Signal Processing
From a practical point of view, implementing large decimation and interpolation factors is not
efficient since high order filters are often needed.
Multirate techniques focus on the upsampler/downsampler to increase the efficiency of DSP systems.
Interchanging of Filtering and Downsampling/Upsampling
To implement a fractional change in the sampling rate it follows that a cascade of an upsampler and a
downsampler should be used. We would like to determine under what conditions a cascade of a factor
of M downsampler and a factor of L upsampler can be interchanged. That is:
nx nyM L nx nyML
It can be shown that this interchange is possible only if M and L are relatively prime. That is, M
and L do not have a common factor that is an integer k > 1.
5-43
XMHXb
M
k
bM
k
MX
MY
0
21
kHM
k
MX
MY
M
k
221
0
nx
nx
ny
ny
nxa
nxb
M
M
zH
MzH
A related case is presented with decimation shown below. In one case a filter follows the down-sampler
(operating at a lower sampling rate) and in the other case a filter precedes the down-sampler (operating at
a higher sampling rate). Both systems are equivalent, as shown in the equations below:
a
M
k
XHM
k
MX
MHY
0
21
M
k
aM
k
MX
MX
0
21
(i)
(ii)
using (ii)
this is just Xa() from (i)
aXHY
5-44
A similar principle applies to interpolation:
nx
nx
ny
ny
nxa
nxb
L zH
LzHL
LHLXLXY a
LXXb
LXLHXLHY b
( ) ( )aX X H
5-45
Polyphase Decomposition
Representing a sequence (or impulse response) as M subsequences, each consisting of every M th value
of successively delayed versions of the sequence.
This approach can lead to efficient implementation of filters
otherwise
of multipleinteger
0
Mnknhnhk
1
0
M
k
k
h n h n k
nMhknMhne kk
These are the “polyphase components” of h(n)
Note: Decomposition, not filtering operation!
5-46
Example of a polyphase decomposition with
M = 3An equivalent representation is shown below.
Note these are not filter realizations, but rather,
they show how a filter can be decomposed into
parallel filters.
)(0 ne
)(1 ne
)(2 ne
n
n
nn
)(nh
5-47
Polyphase Implementation of Decimation Filters
nx
)(nwnMy w n
M zH
1
0
M
k
kM
k zzEzH
filter-then-decimate
decimate-then-filter
nMhknMhne kk
5-48
Polyphase Implementation of Interpolation Filters
nx
( )n
w x nL
y nL zH
filter-then-interpolate
interpolate-then-filter
5-49
5.7 Digital Processing of Analog Signals
All previous signal conversions were ideal as described in the diagram below:
However, practical considerations reveal the following sources of error:
• continuous-time signals are not bandlimited
• ideal filters cannot be realized
• ideal C/D and D/C converters cannot be realized
The way in which signals are converted in actuality is shown below
5-50
A/D and D/A converters have non-ideal behavior that impacts the entire system.
Of course, the general approach is to minimize the sampling rate. Why?
Note that even for ideal C/D converters an anti-aliasing filter is needed in order to band-limit the input signal.
jHaa
cc
T
T
TTH
Heff
,0
,Note that:
caaeff HHH ,
But with the anti-aliasing filter, this becomes
With this approach, sharp cutoff anti-aliasing LPF is required.
We would like to avoid the cost of such filters by using DSP
techniques.
)(aaH
)(aaH
c
)(nh
c
highest
frequency
component
after Haa()
5-51
Consider the following system:
is the highest frequency component to be retained after anti-aliasing filtering
The simple anti-aliasing filter has gradual cutoff but high attenuation at
Here is the approach:
1. Sample much higher than , say
2. Follow by sharp anti-aliasing filter in digital domain
3. Follow by decimation by M
N
NM
NM2N2
5-52
simple
anti-aliasing
filter
sharp
anti-aliasing
filter
C/D
txc
txa
nx̂
nxd
M
NMT
1
)(CX
)(aaX
)(ˆ X
)(dX
5-53
A/D Conversion
The A/D converter is an approximation to the C/D converter as shown below:
Sample
& HoldA/D
TT
txa
tx0
nxBˆ
Binary numbers of B bit accuracy
tho
T
t
n
aoo nTtnTxthtx
We would like the sampling to be
instantaneous and the hold to keep the
sample value constant for T-seconds.
1
5-54
The next step in the conversion process is the quantizer and encoder. Although represented separately,
quantizing and encoding are usually one step
Quantizing – rounding off sample values to the nearest permissible value
Encoding – representing each permissible sample value with a binary digital number
The quantizer is a
nonlinear system
having this as its
input/output
characteristic
input dynamic range
This example is a
mid-tread quantizer
5-55
Analysis of Quantization Errors
Quantization introduces error so that the quantized signal sample is an approximation.
Quantizer
Q nx nxQnx ˆ
where we can define the quantizing error as so that nxnxne ˆ
nx nenxnx ˆ
ne
+
22
ne
The quantizer error is bounded by
B
m
B
m XX
22
21
where
mX
mX2
is the full scale level
is the dynamic range
5-56
A statistical representation is used to represent quantization error. This requires the following assumptions:
• The error sequence is a sample sequence of a stationary random process.
• The error sequence is uncorrelated with the sequence x(n).
• The error sequence is a white noise random process (i. e., error sequence samples are uncorrelated).
• The PDF of the error function is uniform over the range of quantization error.
ep
1
2
2
e
m
B X 2
2
2
222
12
1deee
12
2 222 m
B
e
X
The variance of the error
sequence is effectively the
quantizer “noise power”
in terms of the
quantizer parameters
Noise power alone is not the most useful way of characterizing the impact of quantizing error. The SQNR
is more valuable as a metric:
x
m
m
x
B
e
x XB
X
102
22
102
2
10 log208.1002.6212
log10log10SQNR
“signal power”
5-57
x
m
m
x
B
e
x XB
X
102
22
102
2
10 log208.1002.6212
log10log10SQNR
SQNR increases 6 dB for each additional
bit of quantizer accuracy (i.e., doubling #
of quantizing levels)
2
cos1 2
2
0
2 AdtTA
T
T
ox
TAtx o cos
Adjust the amplitude of x(t) to extend across the dynamic range of the quantizer:
Example: A sinusoid
With signal power:
AX m
2log208.1002.6SQNR 10 B
79.702.6SQNR B
• if x > Xm clipping will occur
• when x gets smaller, the SQNR decreases
• the “optimum” case encompasses the entire signal,
• assuming Gaussian input, “optimal” is approx. by
Xm = 3 x which covers >99% of the signal
x =
√2
A
Can use to determine # bits needed to achieve a desired SQNR.
=> Remember to add a sign bit afterwards (this only accounts for interval [0 Xm], need to quantize over [–Xm Xm]
5-58
D/A Conversion
Previously, for a bandlimited signal sampled at (or above) the Nyquist rate we showed that the ideal
analog reconstruction filter (or interpolation function) has the form
Note that the ideal reconstruction filter extends over all time. Hence, it is noncausal and
physically unrealizable. We now consider some practical reconstruction filters.
t
tth
T
Tr
sin
T
F
TT
H
S
r
,0
22,
hr(t)
t
T
rH
T T
5-59
Zero-Order Hold
The simplest reconstruction filter is implemented by holding the sample value as constant
over the length of the sampling period T. Note that this is the same approach used for A/D
conversion.
hZOH(t)Lowpass
smoothing
filter
digital
input
signal
analog
output
signal
where
otherwise,0
0,1ZOH
Ttth
thZOH
T
t
sharp transition creates
undesirable high frequency
components, LPF to remove
TntnTnTxtx )1()()(ˆ
or
5-60
First-Order Hold
First-order hold approximates x(t) by straight-line segments which have a slope that is determined
by the current sample x(nT) and the previous sample x(nTT).
hFOH(t)Lowpass
smoothing
filter
digital
input
signal
analog
output
signal
where
otherwise,0
2,1
0,1
FOH TtTT
t
TtT
t
th
or
TntnTnTtT
TnTxnTxnTxtx )1()(
)()()()(ˆ
while not as bad as ZOH,
FOH still requires LPF
because it generates undesirable
high frequency components
5-61
Linear Interpolation with Delay
This technique linearly extrapolates the next sample based on the samples x(nT) and x(nTT).
A one sample delay is required to avoid jumps at the sample points.
hLID(t)Lowpass
smoothing
filter
digital
input
signal
analog
output
signal
where
otherwise,0
2,2
0,
LID TtTT
t
TtT
t
th
or
TntnTnTtT
TnTxnTxTnTxtx )1()(
)()()()(ˆ
Note that all non-ideal reconstruction filters must be followed by LPF to remove the
distortions they induce.