Topic 6 Sampling Rate Conversion (std).pdf

Topic 6
Sampling Rate Conversion
1

Introduction
 It is sometimes necessary to change the
sampling rate of a signal.
 For example, in telecommunications, there
are many types of data being transmitted and
received (e.g. facsimile, voice, video etc) and
we need to process the signals at different
rates.
 The process of converting a signal from one
rate to a different rate is called sampling rate
conversion.
 There are two ways to perform sampling rate
conversion.
2

Introduction
 First method: Convert the digital signal back to
analog using a DAC and resample at the new rate.
◦ Advantage – Your new sampling rate can be arbitrary
and does not need to have any relationship with the
old sampling rate.
◦ Disadvantage – Signal distortion caused by signal
reconstruction in the DAC and quantization caused by
the ADC.
 Second method: Sampling rate conversion done
completely in the digital domain.
◦ This avoids the extra DAC and ADC steps from the
first method, thus you don’t introduce more errors.
3

Sampling Rate Reduction -
Downsampling
 To reduce the sampling rate, we define the
new sampling rate as
where M is an integer.
 The downsampled sequence is therefore:
 Essentially, we are taking every M-th
sample only from the original sequence,
x[n], and removing all other samples in
between.
4
s
s MT
T 
'
)
(
]
[
]
[ s
a
d nMT
x
nM
x
n
x 


Sampling Rate Reduction -
Downsampling
 Time-domain analysis
5
M

]
[n
x ]
[n
xd
s
T
2
2

M
s
T

Frequency domain analysis -
Preamble
 First you need to understand the relationship
between the “analog” frequency, , and the
“digital frequency”, .
  is related to  as follows:
 This simply means that the we are normalizing
(or mapping)  = s in X(j) to  = 2 in
X(ej).
6
s
s
s
s T
or
T




 








2
2

Downsampling – Frequency
domain analysis
 After downsampling, Xd(ej) is composed
of infinite copies of Xa(j) which is
frequency scaled through
=Ts= MTs and shifted by integer
multiples of 2/Ts= 2/MTs.
7

domain analysis
 Refer to Appendix I.
 Fig. 4.21 (d) shows the DTFT of the
downsampled signal plotted as a function
of the normalized (“digital”) frequency,
=Ts.
 Compared to before downsampling, the
FT is “stretched”.
 This is because 2 no longer represents
the original sampling frequency, s, but
the new sampling frequency, s/M.
8

domain analysis
 Figure 4.21(e) plots the FT as a function
of continuous (“analog”) frequency.
 The two plots, 4.21(d) and 4.21(e) are
identical, except for the scaling of the
frequency axis through =Ts.
9

domain analysis
 Question: Since you are reducing the
sampling rate when performing
downsampling, what could be a potential
problem?
Hint: Recall the Nyquist Sampling
Theorem.
10

domain analysis
 If the sampling rate is reduced too much, aliasing may
occur.
 In general, to avoid aliasing when downsampling, M
must be chosen such that:
 Alternatively, if the application can tolerate removing
some frequency components, an ideal LPF can be
applied to limit the bandwidth of the signal before
downsampling.
 Appendix II shows what happens when aliasing
occurs during downsampling.
11
M
f
f
or
f
f
M
s
s








 0
0
0
0
2

domain analysis
 The figure below shows a general system for
sampling rate reduction, called a decimator.
 The LPF with cutoff /M is to bandlimit the signal to
avoid aliasing.
12
s
MT

2
s
MT

4
... ...
'
s
T

0
s
MT

4

s
MT

2

)
( 
j
d e
X
s
s MT
T
1
'
1

s
MT

]
[n
x ]
[n
xd
M

Lowpass filter
gain=1
cutoff=/M
... ...
s
T

0
s
T
1
s
T

2
s
T

4
s
T

4

s
T

2

)
( 
j
e
X
Decimator
M


0
M

 


…
…

Sampling Rate Increase -
Upsampling
 To increase the sampling rate, we define the new
sampling rate as
where L is an integer.
 The upsampled sequence is:
 The upsampling process consists of an expander
and ideal LPF.
 The output of the expander is
 Here, we are spacing out the samples from x[n] by
L, and putting L-1 zeros in between.
13
L
T
T s
s /
'


 



otherwise
L
L
n
L
n
x
n
xe
,
0
,...
2
,
,
0
],
/
[
]
[
)
/
(
]
/
[
]
[ L
nT
x
L
n
x
n
x s
a
i 


Sampling Rate Increase -
Upsampling
 Time-domain analysis of output of
expander.
14
L

]
[n
x ]
[n
xe
2

L
s
T
0 1
n
]
0
[
x ]
1
[
x
]
2
[
x
]
3
[
x ]
4
[
x
L L
2 L
3 L
4
0 n

Upsampling – Frequency domain
analysis
 After upsampling, Xe(ej) is a frequency-scaled
version of X(ej) i.e. it is compressed in frequency
by a factor of L. (See Appendix III)
 To obtain Xi(ej), Xe(ej) needs to be corrected in
two ways:
◦ Amplitude-scaling to 1/Ts= L/Ts.
◦ Removing frequency scaled images Xa(j) except at
integer multiples of 2.
 This is achieved by applying an ideal LPF after
the expander with a gain of L and a cutoff at  /L.
 The output of the ideal LPF, xi[n], will be related
to the original x[n] by:
15
)
/
(
)
'
(
]
[ L
nTs
x
nTs
x
n
x a
a
i 


Upsampling – Frequency domain
analysis
16
]
[n
x ]
[n
xi
L

Lowpass filter
gain=L
cutoff=/L
]
[n
xe
... ...

0
s
T
/
1

2
)
( 
j
e
X

 2 
 
Spectrum is scaled
or compressed
... ...

0
s
T
/
1
L

2
2
)
( 
L
e
X j
e

L

 2
L

L

 


2
4
L
... ...

0
s
s T
L
T /
'
/
1 
L

2
)
( 
j
i e
X
L

 2
L

L

 


2
4
L
 The figure below shows a general system for
sampling rate increase, called an
interpolator.

Upsampling – Time domain
analysis
 In the time domain, the LPF performs
interpolation to fill in the missing
samples.
17
]
[n
x ]
[n
xi
L

Lowpass filter
gain=L
cutoff=/L
s
T
0 1
n
]
0
[
x ]
1
[
x
]
2
[
x
]
3
[
x ]
4
[
x
L L
2 L
3 L
4
0 n
]
[n
xe
]
0
[
x ]
1
[
x
]
2
[
x
]
3
[
x ]
4
[
x
L L
2 L
3 L
4
0 n

Sampling Rate Change by a
Rational factor
 To change the sampling rate by a rational
factor, L/M, we cascade the interpolator and
decimator as follows.
 This can be simplified to:
 The filter cutoff is chosen to be the minimum
between  /L and  /M to avoid aliasing.
18
]
[n
x ]
[
~ n
xd
L

Lowpass filter
gain=L
cutoff=/L
M

Lowpass filter
gain=1
cutoff=/M
]
[n
xe ]
[n
xi ]
[
~ n
xi
]
[n
x ]
[
~ n
xd
L

Lowpass filter
gain=L
cutoff=min(/L, /M)
M

]
[n
xe ]
[
~ n
xi

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