Chapter 10. Sampling and Reconstruction
10.1. Signal Sampling*
Introduction
Digital computers can process discrete time signals using extremely flexible and powerful algorithms. However, most signals of interest are continuous time signals, which is how data almost always appears in nature. This module introduces the concepts behind converting continuous time signals into discrete time signals through a process called sampling.
Sampling
Sampling a continuous time signal produces a discrete time signal by selecting the values of the continuous time signal at evenly spaced points in time. Thus, sampling a continuous time signal
x
with sampling period
T
s
gives the discrete time signal
x
s
defined by 
The sampling angular frequency is then given by
ω
s
= 2π / T
s
.
It should be intuitively clear that multiple continuous time signals sampled at the same rate can produce the same discrete time signal since uncountably many continuous time functions could be constructed that connect the points on the graph of any discrete time function. Thus, sampling at a given rate does not result in an injective relationship. Hence, sampling is, in general, not invertible.
Example 10.1.
For instance, consider the signals x,y defined by
and their sampled versions x S ,y s with sampling period T s = π / 2
Notice that since
it follows that
Hence, x and y provide an example of distinct functions with the same sampled versions at a specific sampling rate.
It is also useful to consider the relationship between the frequency domain representations of the continuous time function and its sampled versions. Consider a signal
x
sampled with sampling period
T
s
to produce the discrete time signal 
. The spectrum
X
s
(ω) for
ω ∈ [ – π,π) of
x
s
is given by
Using the continuous time Fourier transform, 
can be represented as
Thus, the unit sampling period version of 
, which is 
can be represented as
This is algebraically equivalent to the representation
which reduces by periodicity of complex exponentials to
Hence, it follows that
Noting that the above expression contains a Fourier series and inverse Fourier series pair, it follows that
Hence, the spectrum of the sampled signal is, intuitively, the scaled sum of an infinite number of shifted and time scaled copies of original signal spectrum. Aliasing, which will be discussed in depth in later modules, occurs when these shifted spectrum copies overlap and sum together. Note that when the original signal
x
is bandlimited to 
no overlap occurs, so each period of the sampled signal spectrum has the same form as the orignal signal spectrum. This suggest that if we sample a bandlimited signal at a sufficiently high sampling rate, we can recover it from its samples as will be further described in the modules on the Nyquist-Shannon sampling theorem and on perfect reconstruction.
Sampling Summary
Sampling a continuous time signal produces a discrete time signal by selecting the values of the continuous time signal at equally spaced points in time. However, we have shown that this relationship is not injective as multiple continuous time signals can be sampled at the same rate to produce the same discrete time signal. This is related to a phenomenon called aliasing which will be discussed in later modules. Consequently, the sampling process is not, in general, invertible. Nevertheless, as will be shown in the module concerning reconstruction, the continuous time signal can be recovered from its sampled version if some additional assumptions hold.
10.2. Sampling Theorem*
Introduction
With the introduction of the concept of signal sampling, which produces a discrete time signal by selecting the values of the continuous time signal at evenly spaced points in time, it is now possible to discuss one of the most important results in signal processing, the Nyquist-Shannon sampling theorem. Often simply called the sampling theorem, this theorem concerns signals, known as bandlimited signals, with spectra that are zero for all frequencies with absolute value greater than or equal to a certain level. The theorem implies that there is a sufficiently high sampling rate at which a bandlimited signal can be recovered exactly from its samples, which is an important step in the processing of continuous time signals using the tools of discrete time signal processing.
Nyquist-Shannon Sampling Theorem
Statement of the Sampling Theorem
The Nyquist-Shannon sampling theorem concerns signals with continuous time Fourier transforms that are only nonzero on the interval ( – B,B) for some constant B . Such a function is said to be bandlimited to ( – B,B). Essentially, the sampling theorem has already been implicitly introduced in the previous module concerning sampling. Given a continuous time signals x with continuous time Fourier transform X , recall that the spectrum X s of sampled signal x s with sampling period T s is given by
It had previously been noted that if
x
is bandlimited to 
, the period of
X
s
centered about the origin has the same form as
X
scaled in frequency since no aliasing occurs. This is illustrated in Figure 10.1. Hence, if any two 
bandlimited continuous time signals sampled to the same signal, they would have the same continuous time Fourier transform and thus be identical. Thus, for each discrete time signal there is a unique 
bandlimited continuous time signal that samples to the discrete time signal with sampling period
T
s
. Therefore, this 
bandlimited signal can be found from the samples by inverting this bijection.
This is the essence of the sampling theorem. More formally, the sampling theorem states the following. If a signal x is bandlimited to ( – B,B), it is completely determined by its samples with sampling rate ω s = 2B . That is to say, x can be reconstructed exactly from its samples x s with sampling rate ω s = 2B . The angular frequency 2B is often called the angular Nyquist rate. Equivalently, this can be stated in terms of the sampling period T s = 2π / ω s . If a signal x is bandlimited to ( – B,B), it is completely determined by its samples with sampling period T s = π / B . That is to say, x can be reconstructed exactly from its samples x s with sampling period T s .
Figure 10.1.
Proof of the Sampling Theorem
The above discussion has already shown the sampling theorem in an informal and intuitive way that could easily be refined into a formal proof. However, the original proof of the sampling theorem, which will be given here, provides the interesting observation that the samples of a signal with period
T
s
provide Fourier series coefficients for the original signal spectrum on 
.
Let
x
be a 
bandlimited signal and
x
s
be its samples with sampling period
T
s
. We can represent
x
in terms of its spectrum
X
using the inverse continuous time Fourier transfrom and the fact that
x
is bandlimited. The result is
This representation of x may then be sampled with sampling period T s to produce
Noticing that this indicates that
x
s
(n) is the
n th
continuous time Fourier series coefficient for
X(ω) on the interval 
, it is shown that the samples determine the original spectrum
X(ω) and, by extension, the original signal itself.
Perfect Reconstruction
Another way to show the sampling theorem is to derive the reconstruction formula that gives the original signal 
from its samples
x
s
with sampling period
T
s
, provided
x
is bandlimited to 
. This is done in the module on perfect reconstruction. However, the result, known as the Whittaker-Shannon reconstruction formula, will be stated here. If the requisite conditions hold, then the perfect reconstruction is given by
where the sinc function is defined as
From this, it is clear that the set
forms an orthogonal basis for the set of 
bandlimited signals, where the coefficients of a 
signal in this basis are its samples with sampling period
T
s
.
Practical Implications
Discrete Time Processing of Continuous Time Signals
The Nyquist-Shannon Sampling Theorem and the Whittaker-Shannon Reconstruction formula enable discrete time processing of continuous time signals. Because any linear time invariant filter performs a multiplication in the frequency domain, the result of applying a linear time invariant filter to a bandlimited signal is an output signal with the same bandlimit. Since sampling a bandlimited continuous time signal above the Nyquist rate produces a discrete time signal with a spectrum of the same form as the original spectrum, a discrete time filter could modify the samples spectrum and perfectly reconstruct the output to produce the same result as a continuous time filter. This allows the use of digital computing power and flexibility to be leveraged in continuous time signal processing as well. This is more thouroughly described in the final module of this chapter.
Psychoacoustics
The properties of human physiology and psychology often inform design choices in technologies meant for interactin with people. For instance, digital devices dealing with sound use sampling rates related to the frequency range of human vocalizations and the frequency range of human auditory sensativity. Because most of the sounds in human speech concentrate most of their signal energy between 5 Hz and 4 kHz, most telephone systems discard frequencies above 4 kHz and sample at a rate of 8 kHz. Discarding the frequencies greater than or equal to 4 kHz through use of an anti-aliasing filter is important to avoid aliasing, which would negatively impact the quality of the output sound as is described in a later module. Similarly, human hearing is sensitive to frequencies between 20 Hz and 20 kHz. Therefore, sampling rates for general audio waveforms placed on CDs were chosen to be greater than 40 kHz, and all frequency content greater than or equal to some level is discarded. The particular value that was chosen, 44.1 kHz, was selected for other reasons, but the sampling theorem and the range of human hearing provided a lower bound for the range of choices.
Sampling Theorem Summary
The Nyquist-Shannon Sampling Theorem states that a signal bandlimited to 
can be reconstructed exactly from its samples with sampling period
T
s
. The Whittaker-Shannon interpolation formula, which will be further described in the section on perfect reconstruction, provides the reconstruction of the unique 
bandlimited continuous time signal that samples to a given discrete time signal with sampling period
T
s
. This enables discrete time processing of continuous time signals, which has many powerful applications.
10.3. Signal Reconstruction*
Introduction
The sampling process produces a discrete time signal from a continuous time signal by examining the value of the continuous time signal at equally spaced points in time. Reconstruction, also known as interpolation, attempts to perform an opposite process that produces a continuous time signal coinciding with the points of the discrete time signal. Because the sampling process for general sets of signals is not invertible, there are numerous possible reconstructions from a given discrete time signal, each of which would sample to that signal at the appropriate sampling rate. This module will introduce some of these reconstruction schemes.
Reconstruction
Reconstruction Process
The process of reconstruction, also commonly known as interpolation, produces a continuous time signal that would sample to a given discrete time signal at a specific sampling rate. Reconstruction can be mathematically understood by first generating a continuous time impulse train
from the sampled signal
x
s
with sampling period
T
s
and then applying a lowpass filter
G
that satisfies certain conditions to produce an output signal 
. If
G
has impulse response
g
, then the result of the reconstruction process, illustrated in Figure 10.2, is given by the following computation, the final equation of which is used to perform reconstruction in practice.
Figure 10.2.
Reconstruction Filters
In order to guarantee that the reconstructed signal 
samples to the discrete time signal
x
s
from which it was reconstructed using the sampling period
T
s
, the lowpass filter
G
must satisfy certain conditions. These can be expressed well in the time domain in terms of a condition on the impulse response
g
of the lowpass filter
G
. The sufficient condition to be a reconstruction filters that we will require is that, for all
n ∈ Z,
This means that
g
sampled at a rate
T
s
produces a discrete time unit impulse signal.
Therefore, it follows that sampling 
with sampling period
T
s
results in
which is the desired result for reconstruction filters.
Cardinal Basis Splines
Since there are many continuous time signals that sample to a given discrete time signal, additional constraints are required in order to identify a particular one of these. For instance, we might require our reconstruction to yield a spline of a certain degree, which is a signal described in piecewise parts by polynomials not exceeding that degree. Additionally, we might want to guarantee that the function and a certain number of its derivatives are continuous.
This may be accomplished by restricting the result to the span of sets of certain splines, called basis splines or B-splines. Specifically, if a
n th
degree spline with continuous derivatives up to at least order
n – 1 is required, then the desired function for a given
T
s
belongs to the span of 
where
for n ≥ 1 and
Figure 10.3.
However, the basis splines B n do not satisfy the conditions to be a reconstruction filter for n ≥ 2 as is shown in Figure 10.3. Still, the B n are useful in defining the cardinal basis splines, which do satisfy the conditions to be reconstruction filters. If we let b n be the samples of B n on the integers, it turns out that b n has an inverse b n – 1 with respect to the operation of convolution for each n . This is to say that b n – 1 * b n = δ . The cardinal basis spline of order n for reconstruction with sampling period T s is defined as
In order to confirm that this satisfies the condition to be a reconstruction filter, note that
Thus, η n is a valid reconstruction filter. Since η n is an n th degree spline with continuous derivatives up to order n – 1, the result of the reconstruction will be a n th degree spline with continuous derivatives up to order n – 1.
Figure 10.4.
The lowpass filter with impulse response equal to the cardinal basis spline η 0 of order 0 is one of the simplest examples of a reconstruction filter. It simply extends the value of the discrete time signal for half the sampling period to each side of every sample, producing a piecewise constant reconstruction. Thus, the result is discontinuous for all nonconstant discrete time signals.
Likewise, the lowpass filter with impulse response equal to the cardinal basis spline η 1 of order 1 is another of the simplest examples of a reconstruction filter. It simply joins the adjacent samples with a straight line, producing a piecewise linear reconstruction. In this way, the reconstruction is continuous for all possible discrete time signals. However, unless the samples are collinear, the result has discontinuous first derivatives.
In general, similar statements can be made for lowpass filters with impulse responses equal to cardinal basis splines of any order. Using the n th order cardinal basis spline η n , the result is a piecewise degree n polynomial. Furthermore, it has continuous derivatives up to at least order n – 1. However, unless all samples are points on a polynomial of degree at most n , the derivative of order n will be discontinuous.
Reconstructions of the discrete time signal given in Figure 10.5 using several of these filters are shown in Figure 10.6. As the order of the cardinal basis spline increases, notice that the reconstruction approaches that of the infinite order cardinal spline η ∞ , the sinc function. As will be shown in the subsequent section on perfect reconstruction, the filters with impulse response equal to the sinc function play an especially important role in signal processing.
Figure 10.5.
Figure 10.6.
Reconstruction Summary
Reconstruction of a continuous time signal from a discrete time signal can be accomplished through several schemes. However, it is important to note that reconstruction is not the inverse of sampling and only produces one possible continuous time signal that samples to a given discrete time signal. As is covered in the subsequent module, perfect reconstruction of a bandlimited continuous time signal from its sampled version is possible using the Whittaker-Shannon reconstruction formula, which makes use of the ideal lowpass filter and its sinc function impulse response, if the sampling rate is sufficiently high.
10.4. Perfect Reconstruction*
Introduction
If certain additional assumptions about the original signal and sampling rate hold, then the original signal can be recovered exactly from its samples using a particularly important type of filter. More specifically, it will be shown that if a bandlimited signal is sampled at a rate greater than twice its bandlimit, the Whittaker-Shannon reconstruction formula perfectly reconstructs the original signal. This formula makes use of the ideal lowpass filter, which is related to the sinc function. This is extremely useful, as sampled versions of continuous time signals can be filtered using discrete time signal processing, often in a computer. The results may then be reconstructed to produce the same continuous time output as some desired continuous time system.
Perfect Reconstruction
In order to understand the conditions for perfect reconstruction and the filter it employs, consider the following. As a beginning, a sufficient condition under which perfect reconstruction is possible will be discussed. Subsequently, the filter and process used for perfect reconstruction will be detailed.
Recall that the sampled version x s of a continuous time signal x with sampling period T s has a spectrum given by
As before, note that if
x
is bandlimited to 
, meaning that
X
is only nonzero on 
, then each period of
X
s
has the same form as
X
. Thus, we can identify the original spectrum
X
from the spectrum of the samples
X
s
and, by extension, the original signal
x
from its samples
x
s
at rate
T
s
if
x
is bandlimited to 
.
If a signal
x
is bandlimited to ( – B,B), then it is also bandlimited to 
provided that
T
s
< π / B
. Thus, if we ensure that
x
is sampled to
x
s
with sufficiently high sampling angular frequency
ω
s
= 2π / T
s
> 2B
and have a way of identifying the unique 
bandlimited signal corresponding to a discrete time signal at sampling period
T
s
, then
x
s
can be used to reconstruct 
exactly. The frequency 2B
is known as the angular Nyquist rate. Therefore, the condition that the sampling rate
ω
s
= 2π / T
s
> 2B
be greater than the Nyquist rate is a sufficient condition for perfect reconstruction to be possible.
The correct filter must also be known in order to perform perfect reconstruction. The ideal lowpass filter defined by
G(ω) = T
s
(u (ω + π / T
s
) – u (ω – π / T
s
)), which is shown in Figure 10.7, removes all signal content not in the frequency range 
. Therefore, application of this filter to the impulse train 
results in an output bandlimited to 
.
We now only need to confirm that the impulse response g of the filter G satisfies our sufficient condition to be a reconstruction filter. The inverse Fourier transform of G(ω) is
which is shown in Figure 10.7. Hence,
Therefore, the ideal lowpass filter
G
is a valid reconstruction filter. Since it is a valid reconstruction filter and always produces an output that is bandlimited to 
, this filter always produces the unique 
bandlimited signal that samples to a given discrete time sequence at sampling period
T
s
when the impulse train 
is input.
Therefore, we can always reconstruct any 
bandlimited signal from its samples at sampling period
T
s
by the formula
This perfect reconstruction formula is known as the Whittaker-Shannon interpolation formula and is sometimes also called the cardinal series. In fact, the sinc function is the infinite order cardinal basis spline
η
∞
. Consequently, the set 
forms a basis for the vector space of 
bandlimited signals where the signal samples provide the corresponding coefficients. It is a simple exercise to show that this basis is, in fact, an orthogonal basis.
Figure 10.7.
Figure 10.8.
Perfect Reconstruction Summary
This module has shown that bandlimited continuous time signals can be reconstructed exactly from their samples provided that the sampling rate exceeds the Nyquist rate, which is twice the bandlimit. The Whittaker-Shannon reconstruction formula computes this perfect reconstruction using an ideal lowpass filter, with the resulting signal being a sum of shifted sinc functions that are scaled by the sample values. Sampling below the Nyquist rate can lead to aliasing which makes the original signal irrecoverable as is described in the subsequent module. The ability to perfectly reconstruct bandlimited signals has important practical implications for the processing of continuous time signals using the tools of discrete time signal processing.
10.5. Aliasing Phenomena*
Introduction
Through discussion of the Nyquist-Shannon sampling theorem and Whittaker-Shannon reconstruction formula, it has already been shown that a ( – B,B) continuous time signal can be reconstructed from its samples at rate ω s = 2π / T s via the sinc interpolation filter if ω s > 2B . Now, this module will investigate a problematic phenomenon, called aliasing, that can occur if this sufficient condition for perfect reconstruction does not hold. When aliasing occurs the spectrum of the samples has different form than the original signal spectrum, so the samples cannot be used to reconstruct the original signal through Whittaker-Shannon interpolation.
Aliasing
Aliasing occurs when each period of the spectrum of the samples does not have the same form as the spectrum of the original signal. Given a continuous time signals x with continuous time Fourier transform X , recall that the spectrum X s of sampled signal x s with sampling period T s is given by
As has already been mentioned several times, if
x
is bandlimited to 
then each period of
X
s
has the same form as
X
. However, if
x
is not bandlimited to 
, then the 
can overlap and sum together. This is illustrated in Figure 10.9 in which sampling above the Nyquist frequency produces a samples spectrum of the same shape as the original signal, but sampling below the Nyquist frequency produces a samples spectrum with very different shape. Whittaker-Shannon interpolation of each of these sequences produces different results. The low frequencies not affected by the overlap are the same, but there is noise content in the higher frequencies caused by aliasing. Higher frequency energy masquerades as low energy content, a highly undesirable effect.
Figure 10.9.
Unlike when sampling above the Nyquist frequency, sampling below the Nyquist frequency does not yield an injective (one-to-one) function from the ( – B,B) bandlimited continuous time signals to the discrete time signals. Any signal x with spectrum X which overlaps and sums to X s samples to x s . It should be intuitively clear that there are very many ( – B,B) bandlimited signals that sample to a given discrete time signal below the Nyquist frequency, as is demonstrated in Figure 10.10. It is quite easy to construct uncountably infinite families of such signals.
Aliasing obtains it name from the fact that multiple, in fact infinitely many, ( – B,B) bandlimited signals sample to the same discrete sequence if
ω
s
< 2B
. Thus, information about the original signal is lost in this noninvertible process, and these different signals effectively assume the same identity, an “alias”. Hence, under these conditions the Whittaker-Shannon interpolation formula will not produce a perfect reconstruction of the original signal but will instead give the unique 
bandlimited signal that samples to the discrete sequence.
Figure 10.10.
Aliasing Demonstration
Figure 10.11.
Aliasing Summary
Aliasing, essentially the signal processing version of identity theft, occurs when each period of the spectrum of the samples does not have the same form as the spectrum of the original signal. As has been shown, there can be infinitely many ( – B,B) bandlimited signals that sample to a given discrete time signal x s at a rate ω s = 2π / T s < 2B below the Nyquist frequency. However, there is a unique ( – B,B) bandlimited signal that samples to x s , which is given by the Whittaker-Shannon interpolation of x s , at rate ω s ≥ 2B as no aliasing occurs above the Nyquist frequency. Unfortunately, sufficiently high sampling rates cannot always be produced. Aliasing is detrimental to many signal processing applications, so in order to process continuous time signals using discrete time tools, it is often necessary to find ways to avoid it other than increasing the sampling rate. Thus, anti-aliasing filters, are of practical importance.
10.6. Anti-Aliasing Filters*
Introduction
It has been shown that a ( – B,B) bandlimited signal can be perfectly reconstructed from its samples at a rate
ω
s
= 2π / T
s
≥ B
. However, it is not always practically possible to produce sufficiently high sampling rates or to ensure that the input is bandlimited in real situations. Aliasing, which manifests itself as a difference in shape between the periods of the samples signal spectrum and the original spectrum, would occur without any further measures to correct this. Thus, it often becomes necessary to filter out signal energy at frequencies above
ω
s
/ 2 in order to avoid the detrimental effects of aliasing. This is the role of the anti-aliasing filter, a lowpass filter applied before sampling to ensure that the signal is 
bandlimited or at least nearly so.
Anti-Aliasing Filters
Aliasing can occur when a signal with energy at frequencies other that ( – B,B) is sampled at rate ω s < 2B . Thus, when sampling below the Nyquist frequency, it is desirable to remove as much signal energy outside the frequency range ( – B,B) as possible while keeping as much signal energy in the frequency range ( – B,B) as possible. This suggests that the ideal lowpass filter with cutoff frequency ω s / 2 would be the optimal anti-aliasing filter to apply before sampling. While this is true, the ideal lowpass filter can only be approximated in real situations.
In order to demonstrate the importance of anti-aliasing filters, consider the calculation of the error energy between the original signal and its Whittaker-Shannon reconstruction from its samples taken with and without the use of an anti-aliasing filter. Let x be the original signal and y = G x be the anti-alias filtered signal where G is the ideal lowpass filter with cutoff frequency ω s / 2. It is easy to show that the reconstructed spectrum using no anti-aliasing filter is given by
Thus, the reconstruction error spectrum for this case is
Similarly, the reconstructed spectrum using the ideal lowpass anti-aliasing filter is given by
Thus, the reconstruction error spectrum for this case is
Hence, by Parseval's theorem, it follows that 
. Also note that the spectrum of 
is identical to that of the original signal
X
at frequencies 
This is graphically shown in Figure 10.12.
Figure 10.12.
Anti-Aliasing Filters Summary
As can be seen, anti-aliasing filters ensure that the signal is 
bandlimited, or at least nearly so. The optimal anti-aliasing filter would be the ideal lowpass filter with cutoff frequency at
ω
s
/ 2, which would ensure that the original signal spectrum and the reconstructed signal spectrum are equal on the interval 
. However, the ideal lowpass filter is not possible to implement in practice, and approximations must be accepted instead. Anti-aliasing filters are an important component of systems that implement discrete time processing of continuous time signals, as will be shown in the subsequent module.
10.7. Discrete Time Processing of Continuous Time Signals*
Introduction
Digital computers can process discrete time signals using extremely flexible and powerful algorithms. However, most signals of interest are continuous time signals, which is how data almost always appears in nature. Now that the theory supporting methods for generating a discrete time signal from a continuous time signal through sampling and then perfectly reconstructing the original signal from its samples without error has been discussed, it will be shown how this can be applied to implement continuous time, linear time invariant systems using discrete time, linear time invariant systems. This is of key importance to many modern technologies as it allows the power of digital computing to be leveraged for processing of analog signals.
Discrete Time Processing of Continuous Time Signals
Process Structure
With the aim of processing continuous time signals using a discrete time system, we will now examine one of the most common structures of digital signal processing technologies. As an overview of the approach taken, the original continuous time signal x is sampled to a discrete time signal x s in such a way that the periods of the samples spectrum X s is as close as possible in shape to the spectrum of X . Then a discrete time, linear time invariant filter H 2 is applied, which modifies the shape of the samples spectrum X s but cannot increase the bandlimit of X s , to produce another signal y s . This is reconstructed with a suitable reconstruction filter to produce a continuous time output signal y , thus effectively implementing some continuous time system H 1 . This process is illustrated in Figure 10.13, and the spectra are shown for a specific case in Figure 10.14.
Figure 10.13.
Further discussion about each of these steps is necessary, and we will begin by discussing the analog to digital converter, often denoted by ADC or A/D. It is clear that in order to process a continuous time signal using discrete time techniques, we must sample the signal as an initial step. This is essentially the purpose of the ADC, although there are practical issues that which will be discussed later. An ADC takes a continuous time analog signal as input and produces a discrete time digital signal as output, with the ideal infinite precision case corresponding to sampling. As stated by the Nyquist-Shannon Sampling theorem, in order to retain all information about the original signal, we usually wish sample above the Nyquist frequency ω s ≥ 2B where the original signal is bandlimited to ( – B,B). When it is not possible to guarantee this condition, an anti-aliasing filter should be used.
The discrete time filter is where the intentional modifications to the signal information occur. This is commonly done in digital computer software after the signal has been sampled by a hardware ADC and before it is used by a hardware DAC to construct the output. This allows the above setup to be quite flexible in the filter that it implements. If sampling above the Nyquist frequency the. Any modifications that the discrete filter makes to this shape can be passed on to a continuous time signal assuming perfect reconstruction. Consequently, the process described will implement a continuous time, linear time invariant filter. This will be explained in more mathematical detail in the subsequent section. As usual, there are, of course, practical limitations that will be discussed later.
Finally, we will discuss the digital to analog converter, often denoted by DAC or D/A. Since continuous time filters have continuous time inputs and continuous time outputs, we must construct a continuous time signal from our filtered discrete time signal. Assuming that we have sampled a bandlimited at a sufficiently high rate, in the ideal case this would be done using perfect reconstruction through the Whittaker-Shannon interpolation formula. However, there are, once again, practical issues that prevent this from happening that will be discussed later.
Figure 10.14.
Discrete Time Filter
With some initial discussion of the process illustrated in Figure 10.13 complete, the relationship between the continuous time, linear time invariant filter H 1 and the discrete time, linear time invariant filter H 2 can be explored. We will assume the use of ideal, infinite precision ADCs and DACs that perform sampling and perfect reconstruction, respectively, using a sampling rate ω s = 2π / T s ≥ 2B where the input signal x is bandlimited to ( – B,B). Note that these arguments fail if this condition is not met and aliasing occurs. In that case, preapplication of an anti-aliasing filter is necessary for these arguments to hold.
Recall that we have already calculated the spectrum X s of the samples x s given an input x with spectrum X as
Likewise, the spectrum Y s of the samples y s given an output y with spectrum Y is
From the knowledge that 
, it follows that
Because
X
is bandlimited to 
, we may conclude that
More simply stated,
H
2
is 2π
periodic and 
for
ω ∈ [ – π,π).
Given a specific continuous time, linear time invariant filter
H
1
, the above equation solves the system design problem provided we know how to implement
H
2
. The filter
H
2
must be chosen such that it has a frequency response where each period has the same shape as the frequency response of
H
1
on 
. This is illustrated in the frequency responses shown in Figure 10.14.
We might also want to consider the system analysis problem in which a specific discrete time, linear time invariant filter
H
2
is given, and we wish to describe the filter
H
1
. There are many such filters, but we can describe their frequency responses on 
using the above equation. Isolating one period of
H
2(ω) yields
the conclusion that 
for 
. Because
x
was assumed to be bandlimited to ( – π / T,π / T), the value of the frequency response elsewhere is irrelevant.
Practical Considerations
As mentioned before, there are several practical considerations that need to be addressed at each stage of the process shown in Figure 10.13. Some of these will be briefly addressed here, and a more complete model of how discrete time processing of continuous time signals appears in Figure 10.15.
Figure 10.15.
Anti-Aliasing Filter
In reality, we cannot typically guarantee that the input signal will have a specific bandlimit, and sufficiently high sampling rates cannot necessarily be produced. Since it is imperative that the higher frequency components not be allowed to masquerade as lower frequency components through aliasing, anti-aliasing filters with cutoff frequency less than or equal to ω s / 2 must be used before the signal is fed into the ADC. The block diagram in Figure 10.15 reflects this addition.
As described in the previous section, an ideal lowpass filter removing all energy at frequencies above ω s / 2 would be optimal. Of course, this is not achievable, so approximations of the ideal lowpass filter with low gain above ω s / 2 must be accepted. This means that some aliasing is inevitable, but it can be reduced to a mostly insignificant level.
Signal Quantization
In our preceding discussion of discrete time processing of continuous time signals, we had assumed an ideal case in which the ADC performs sampling exactly. However, while an ADC does convert a continuous time signal to a discrete time signal, it also must convert analog values to digital values for use in a digital logic device, a phenomenon called quantization. The ADC subsystem of the block diagram in Figure 10.15 reflects this addition.
The data obtained by the ADC must be stored in finitely many bits inside a digital logic device. Thus, there are only finitely many values that a digital sample can take, specifically 2 N where N is the number of bits, while there are uncountably many values an analog sample can take. Hence something must be lost in the quantization process. The result is that quantization limits both the range and precision of the output of the ADC. Both are finite, and improving one at constant number of bits requires sacrificing quality in the other.
Filter Implementability
In real world circumstances, if the input signal is a function of time, the future values of the signal cannot be used to calculate the output. Thus, the digital filter H 2 and the overall system H 1 must be causal. The filter annotation in Figure 10.15 reflects this addition. If the desired system is not causal but has impulse response equal to zero before some time t 0 , a delay can be introduced to make it causal. However, if this delay is excessive or the impulse response has infinite length, a windowing scheme becomes necessary in order to practically solve the problem. Multiplying by a window to decrease the length of the impulse response can reduce the necessary delay and decrease computational requirements.
Take, for instance the case of the ideal lowpass filter. It is acausal and infinite in length in both directions. Thus, we must satisfy ourselves with an approximation. One might suggest that these approximations could be achieved by truncating the sinc impulse response of the lowpass filter at one of its zeros, effectively windowing it with a rectangular pulse. However, doing so would produce poor results in the frequency domain as the resulting convolution would significantly spread the signal energy. Other windowing functions, of which there are many, spread the signal less in the frequency domain and are thus much more useful for producing these approximations.
Anti-Imaging Filter
In our preceding discussion of discrete time processing of continuous time signals, we had assumed an ideal case in which the DAC performs perfect reconstruction. However, when considering practical matters, it is important to remember that the sinc function, which is used for Whittaker-Shannon interpolation, is infinite in length and acausal. Hence, it would be impossible for an DAC to implement perfect reconstruction.
Instead, the DAC implements a causal zero order hold or other simple reconstruction scheme with respect to the sampling rate
ω
s
used by the ADC. However, doing so will result in a function that is not bandlimited to 
. Therefore, an additional lowpass filter, called an anti-imaging filter, must be applied to the output. The process illustrated in Figure 10.15 reflects these additions. The anti-imaging filter attempts to bandlimit the signal to 
, so an ideal lowpass filter would be optimal. However, as has already been stated, this is not possible. Therefore, approximations of the ideal lowpass filter with low gain above
ω
s
/ 2 must be accepted. The anti-imaging filter typically has the same characteristics as the anti-aliasing filter.
Discrete Time Processing of Continuous Time Signals Summary
As has been show, the sampling and reconstruction can be used to implement continuous time systems using discrete time systems, which is very powerful due to the versatility, flexibility, and speed of digital computers. However, there are a large number of practical considerations that must be taken into account when attempting to accomplish this, including quantization noise and anti-aliasing in the analog to digital converter, filter implementability in the discrete time filter, and reconstruction windowing and associated issues in the digital to analog converter. Many modern technologies address these issues and make use of this process.