Transcript Document

Discrete-time Processing of
Continuous-time Signals
(cf. Oppenheim, 1999)
A major application of discrete-time systems is in the
processing of continuous-time signals.

The overall system is equivalent to a continuous-time
system, since it transforms the continuous-time input signal
xs(t) into the continuous time signal yr(t).
 Question: what is this equivalent system?

Ideal discrete-to-continuous (D/C)
converter
Ideal reconstruction filter
The ideal
reconstruction filter is
a continuous-time filter,
with the frequency
response being Hr(j)
and impulse response
hr(t).
hr (t )  sin(t / T ) /(t / T )
From now on, we use
 to represent the
transform domain of
continuous Fourier
transform.
Continuous to discrete (C/D) converter
st  

 t  nT 
continuous
F. T.
k  
where s = 2/T
xs t   x(t ) s(t ) 

2
S  j 
T
 xc (t ) t  nT 
k  

   k 
k  
Time domain
multiplication
s
C/D converter
1
X c ( j )  S ( j )
We have X s ( j) 
2
Frequency domain
convolution
Hence, the continuous Fourier transforms of xs(t) consists of
periodically repeated copies of the Fourier transform of xc(t).

1 
X s  j   X c  j   k s  where s = 2/T
T k  

Review of Nyquist sampling theorem:
Aliasing effect: If s > 2N, the copies of Xc(j) overlap, where N
is the highest nonzero frequency component of Xc(j). N is
referred to as the Nyquist frequency.

Ideal C/D converter


In the above, we characterize the relationship of xs(t) and
xc(t) in the continuous F.T. domain.
From another point of view, Xs(j) can be represented as the
linear combination of a serious of complex exponentials:
X s  j 
since xs t  


 jTn


x
nT
e
c
n  

 x nT  t  nT 
n  
c
If x(nT)  x[n], its DTFT is
X (e jw ) 

 jwn
x
[
n
]
e


n  

 jwn
x
(
nT
)
e

n  
Input-output relationship of C/D
converter

Combining these properties, we have the relationship
between the continuous F.T. and DTFT of the sampled
signal:
jw
jT
 
X s  j   X e
w  T

X e

where
X c  j : represent continuous F.T.
X e jw : represent DTFT
Thus, we have the input-output relationship of C/D
converter


 

 
Xe
jw
 
  w 2k  
1
2k   1
  X c  j  
    X c  j 
 
T k   
T   T k    T
T 
Combine C/D, discrete-time system,
and D/C


Consider again the discrete-time processing of continuous
signals
Let H(ejw) be the frequency response of the discrete-time
system in the above diagram. Since Y(ejw) = H(ejw)X(ejw)



 
Y e jT  H e jT X e jT

D/C converter revisited


An ideal low-pass filter Hr(j) that has a cut-off frequency
c= s/2 = /T and gain T is used for reconstructing the
continuous signal.
Frequency domain of D/C converter: (Hr(j) is its
frequency response)

Yr  j  H r  jY e

jT



TY e jT ,    / T

otherwise
 0,
Remember that the corresponding impulse response is a
sinc function, and the reconstructed signal is
sin  t  nT  / T 
yr t    yn
 t  nT  / T
n  

Input-output Relationship for Discrete-time
processing of continuous-time signals
Assumption:
If Xc(j) is band limited with Xc(j) = 0 for ||>/T, then

1
jT
X c  j   X e
T
and hence

Yr  j   H r  j Y e


jT



 

TH e jT X e jT ,    / T

0,
otherwise

H e jT X c  j ,    / T

0,
otherwise

Effective Frequency Response
for Discrete-time processing of
continuous-time signals
So, if Xc(j) is band limited with Xc(j) = 0 for ||>/T, we have
the following effective response for the entire system:
Yr  j   H eff  j X c  j ,


jT

H
e
,  /T

where H eff  j   

 /T
 0,
Effective Frequency Response
It is important to emphasize that the LTI behavior of the
system depends on two factors:
 First, the discrete-time system is LTI.
 Second, the input signal is band-limited, and the
sampling rate is high enough so that any aliased
components are removed.

Discrete-time processing of continuoustime signals


If we are given a desired continuous-time system with
band-limited frequency response Hc(j), and we want to
implement it by discrete-time processing,
We can choose appropriate T and discrete-time filter
satisfying H(ejw) = Hc(jw/T) to synthesize the continuous
response Hc(j).
Time-domain behavior of discrete-time
processing of continuous-time signals

In time domain
Since H(ejw) = Hc(jw/T) |w| < 
 In addition, Hc(j) = 0, ||>/T (band limited)
 We have the impulse invariance property, h[n]=Thc(nT),
i.e., the impulse response of the discrete-time system is
a scaled, sampled version of the continuous inpulse
response hc(t).
Remark: It is because that if h[n]  Thc (nT)

w 2k
H (e )   H c ( j ( 
))
T
T
k  
w
jw
and thus H (e )  H c ( j ) | w |  since band limited.
T

jw
Continuous-time processing of
discrete-time signals



On the other hand, we can also consider to process discretetime signal with continuous-time filters.
Cascading D/C, continuous-time system, and C/D.
From the definition of the ideal D/C converter, Xc(j) and
therefore also Yc(j), will necessarily be zero for ||>/T.
Continuous-time processing of
discrete-time signals
 
Ye
jw
1  w
 Yc  j 
T  T
| w | 
Yc  j  H c  jX c  j

X c  j  TX e jT


|  |  / T
|  |  / T
Hence, we have the effective system of the continuoustime processing of discrete-time signals to be
 
H eff e
jw
 w
 H c  j ,
 T
w 
Changing the Sampling rate using
discrete-time processing

downsampling; sampling rate compressor;
xd [n]  xnM 
Frequency domain of
downsampling

Since this is a ‘re-sampling’ process. Remember that,
from continuous-time sampling of x[n]=xc(nT), we
have

 
X e

jw
1
 w 2k 
  X c  j( 
)
T k   T
T 
Similarly, for the down-sampled signal xd[m]=xc(mT’),
(where T’ = MT), we have
 
X d e jw
1 
 w 2r 
  X c  j( 
)
T ' r   T ' T ' 
Frequency domain of
downsampling

We are interested in the relation between X(ejw) and
Xd(ejw). Let’s represent r as r = i + kM, where 0  i 
M1, (i.e., r  i (mod M)). Then

1
w 2r 

jw
X d e  
X c  j(

)

MT r   MT MT 
1

M
1 
w 2k 2i 

X c  j(


)


T
MT 
 MT
i  0 T k  
M 1
1 
 w  2i 2k 
  X c  j(

)   X (e j ( w2i ) / M )
T k  
MT
T 
Frequency domain of
downsampling

Therefore, the downsampling can be treated as a ‘resampling’ process. It s frequency domain relationship is
similar to that of the D/C converter as:
 
Xd e


jw
1

M
M 1

j w 2i  / M
X
e


i 0
This is equivalent to compositing M copies of the of X(ejw),
frequency scaled by M and shifted by inter multiples of 2.
The aliasing can be avoided by ensuring that X(ejw) is
bandlimited as
 
X e jw  0 for wN | w |  , and 2/M  2wN
Example of downsampling in the
Frequency domain (without aliasing)
Sampling with a sufficiently large rate which avoids aliasing
Example of downsampling in the
Frequency domain (without aliasing)
Downsampling by 2 (M=2)
Downsampling with prefiltering to
avoid aliasing (decimation)



From the above, the DTFT of the down-sampled signal is the
superposition of M shifted/scaled versions of the DTFT of the original
signal.
To avoid aliasing, we need wN</M, where wN is the highest
frequency of the discrete-time signal x[n].
Hence, downsampling is usually accompanied with a pre-low-pass
filtering, and a low-pass filter followed by down-sampling is usually
called a decimator, and termed the process as decimation.
Up-sampling

Upsampling; sampling rate expander.
:
or equivalently,
 xn / L , n  0 ,  L ,  2 L , ...
xe n  
otherwise
 0,
xe n 
In frequency domain:

 xk  n  kL
k  
    ( x[k ] [n  kL])e
X e e jw 


n   k  
 jwn


 
 jwLk
jwL
(
x
[
k
]
e
)

X
e

k  
Example of up-sampling
Upsampling in the frequency domain
Up-sampling with post low-pass
filtering


Similar to the case of D/C converter, upsampoling is usually
companied with a post low-pass filter with cutoff frequency
/L and gain L, to reconstruct the sequence.
A low-pass filter followed by up-sampling is called an
interpolator, and the whole process is called interpolation.
Example of up-sampling followed by
low-pass filtering
Applying low-pass filtering
Interpolation

Similar to the ideal D/C converter,

If we choose an ideal lowpass filter with cutoff frequency
/L and gain L, its impulse response is
sinn / L
hi n 
n / L

Hence
 

xi n  xe n hi n  
xk  n  kL  hi n


 k  


sin n  kL  / L 

xk 
 n  kL  / L
k  


Its an interpolation of the
discrete sequence x[k]
Sampling rate conversion by a
non-integer rational factor

By combining the decimation and interpolation, we
can change the sampling rate of a sequence.


Changing the sampling rate by a non-integer factor T’ =
TM/L.
Eg., L=100 and M=101, then T’ = 1.01T.
Changing the Sampling rate using
discrete-time processing

Since the interpolation and decimation filters are
in cascade, they can be combined as shown above.
Digital Processing of Analog Signals

Pre-filtering to avoid aliasing



It is generally desirable to minimize the sampling rate.
Eg., in processing speech signals, where often only the lowfrequency band up to about 3-4k Hz is required, even though
the speech signal may have significant frequency content in the
4k to 20k Hz range.
Also, even if the signal is naturally bandlimited, wideband
additive noise may fll in the higher frequency range, and as a
result of sampling. These noise components would be aliased
into the low frequency band.
Over-sampled A/D conversion


The anti-aliasing filter is an analog filter. However, in
applications involving powerful, but inexpensive, digital
processors, these continuous-time filters may account for a
major part of the cost of a system.
Instead, we first apply a very simple anti-aliasing filter that
has a gradual cutoff (instead of a sharp cutoff) with
significant attenuation at MN. Next, implement the C/D
conversion at the sampling rate higher than 2MN. After
that, sampling rate reduction by a factor of M that includes
sharp anti-aliasing filtering is implemented in the discretetime domain.
Using over-sampled A/D conversion to
simplify a continuous-time anti-aliasing filter
Example of over-sampled A/D
conversion
Example of over-sampled A/D
conversion
Sample and hold
x0 (t ) 

 x[n]h (t  nT )
n  
0
1 0  t  T
h0 (t )  
0 otherwise
Example of sample and hold
Quantizer (Quantization)

The real-valued signal has to be stored as a code for
digital processing. This step is called quantization.
xˆ[n]  Q( x[n])

The quantizer is a nonlinear system.

Typically, we apply two’s complement code for representation.
Quantizer (Quantization)
Quantizer (Quantization)

In general, if we have a (B+1)-bit binary two’s complement
fraction of the form:
a0 a1a2 ...aB

0
1
2
B

a
2

a
2

a
2

...

a
2
then its value is
0
1
2
B


  2 X m / 2B1  X m / 2B
The step size of the quantizer is
where Xm is the full scale level of the A/D converter.
The numerical relationship beween the code words and the
quantizer samples is
xˆ[n]  X m xˆB [n]
Example of quantization
Analysis of quantization errors


Quantization error e[n]  xˆ[n]  x[n]
In general, for a (B+1)-bit quantizer with step size , the
quantization error satisfies that
  / 2  e[n]   / 2
when

( X m   / 2)  x[n]  ( X m   / 2)
If x[n] is outside this range, then the quantization error is
larger in magnitude than /2, and such samples are saided
to be clipped.
Analysis of quantization errors


Analyzing the quantization by introducing an error source
and linearizing the system:
The model is equivalent to quantizer if we know e[n].
Assumptions about e[n]





e[n] is a sample sequence of a stationary random process.
e[n] is uncorrelated with the sequence x[n].
The random variables of the error process e[n] are
uncorrelated; i.e., the error is a white-noise process.
The probability distribution of the error process is uniform
over the range of quantization error (i.e., without being
clipped).
The assumptions would not be justified. However, when
the signal is a complicated signal (such as speech or
music), the assumptions are more realistic.

Experiments have shown that, as the signal becomes more
complicated, the measured correlation between the signal and the
quantization error decreases, and the error also becomes
uncorrelated.
Example of quantization error
original signal
3-bit quantization result
3-bit quantization error
Example of quantization error
8-bit quantization error

In a heuristic sense, the assumptions of the statistical
model appear to be valid if the signal is sufficiently complex
and the quantization steps are sufficiently small, so that the
amplitude of the signal is likely to traverse many
quantization steps from sample to sample.
Quantization error analysis
  / 2  e[n]   / 2

e[n] is a white noise sequence. The probability density
function of e[n] is
Quantization error analysis

The mean value of e[n] is zero, and its variance is
/2
2
1

 e2   e 2 de 

12
 / 2

Xm
Since   B
2
For a (B+1)-bit quantizer with full-scale value Xm, the noise
variance, or power, is
2 B
2
2
X
m
 e2 
12
Quantization error analysis

A common measure of the amount of degradation of a signal
by additive noise is the signal-to-noise ratio (SNR), defined as
the ratio of signal variance (power) to noise variance.
Expressed in decibels (dB), the SNR of a (B+1)-bit quantizer is
  x2 
 12  2 2 B  x2 

SNR  10 log10  2   10 log10 
2
Xm
e 


 Xm 

 6.02B  10.8  20 log10 
 x 

Hence, the SNR increases approximately 6dB for each bit added
to the world length of the quantized samples.
Quantization error analysis



The equation can be further simplified for analysis. For example,
if the signal amplitude has a Gaussian distribution, only 0.064
percent of the samples would have an amplitude greater than
4x.
Thus to avoid clipping the peaks of the signal (as is assumed in
our statistical model), we might set the gain of filters and
amplifiers preceding the A/D converter so that x = Xm/4. Using
this value of x gives SNR  6 B  1.25dB
For example, obtaining a SNR about 90-96 dB in high-quality
music recording and playback requires 16-bit quantization.

But it should be remembered that such performance is obtained only if
the input signal is carefully matched to the full-scale of the A/D converter.