Chapter 7. Analog to Digital Conversion

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Chapter 7. Analog to Digital Conversion

• Essentials of Communication Systems Engineering
• John G. Proakis and Masoud Salehi

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Chapter 7. Analog to Digital Conversion

• In order to convert an analog signal to a digital
  signal, i.e., a stream of bits,
• three operations must be completed.
• First, the analog signal has to be sampled, so
  that we can obtain a discrete-time
  continuous-valued signal from the analog
  signal.This operation is called sampling.
• Then the sampled values, which can take an
  infinite number of values are quantized, i.e.,
  rounded to a finite number of values. This is
  called the quantization process.
• After quantization, we have a discrete-time,
  discrete-amplitude signal. The third stage in
  analog-to-digital conversion is encoding. In
  encoding, a sequence of bits (ones and zeros) are
  assigned to different outputs of the quantizer.
  Since the possible outputs of the quantizer are
  finite, each sample of the signal can be
  represented by a finite number of bits. For
  instance, if the quantizer has 256 28 possible
  levels, they can be represented by 8 bits.

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7.1 Sampling of Signals and Signal
Reconstruction from Samples
7.1.1 The Sampling Theorem
Figure 7.1 Sampling of signals.
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The (Shannons) Sampling Theorem

• It basically states two facts
• If the signal x(t) is bandlimited to W, i.e., if
  X(f) ? 0 for f ? W, then it is sufficient to
  sample at intervals Ts 1/(2W) recover the exact
  original signal from the samples.
• We may recover the signal x(t) by lowpass
  filtering the samples with the cutoff frequency
  W.
• Proof)

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Sampling Theorem

• Now if Ts gt 1/(2W), then the replicated spectrum
  of x(t) overlaps and reconstruction of the
  original signal is not possible.
• This type of distortion, which results from
  undersampling, is known as aliasing error or
  aliasing distortion.
• However, if Ts ? 1/(2W), no overlap occurs and
  by employing an appropriate filter we can
  reconstruct the original signal.

Figure 7.2 Frequency-domain representation of the
sampled signal.
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Sampling Theorem

• The sampling rate fs 2W is the minimum sampling
  rate at which no aliasing occurs.
• This sampling rate is known as the Nyquist
  sampling rate.
• If sampling is done at the Nyquist rate, then the
  only choice for the reconstruction filter is an
  ideal lowpass filter and W' W 1/(2Ts).
• In practical systems, sampling is done at a rate
  higher than the Nyquist rate.
• This allows for the reconstruction filter to be
  realizable and easier to build.
• In such cases, the distance between two adjacent
  replicated spectra in the frequency domain, i.e.,
  (1/Ts - W) - W fs - 2W, is known as the guard
  band.
• Therefore, in systems with a guard band, we have
  fs 2W WG, where W is the bandwidth of the
  signal, WG is the guard band, and fs is the
  sampling frequency.

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7.1.2 (Analog) Pulse Modulation

• Pulse Amplitude Modulation (PAM)
• Sample and hold
• Instantaneous sampling
• Lengthening(T)
• Pulse Duration/Width Modulation (PDM/PWM)
• Samples of the message signal are used to vary
  the duration(width) of the individual pulses in
  the carrier
• Pulse Position Modulation (PPM)
• The position of a pulse relative to its
  unmodulated time of occurrence is varied in
  accordance with the message signal

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(Analog) Pulse Modulation Demodulation? ????

• PDM(PWM)

• PPM

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7.2 QUANTIZATION

• After sampling, we have a discrete-time signal,
  i.e., a signal with values at integer multiples
  of Ts.
• The amplitudes of these signals are still
  continuous, however.
• Transmission of real numbers requires an infinite
  number of bits, since generally the base 2
  representation of real numbers has infinite
  length.
• After sampling, we will use quantization, in
  which the amplitude becomes discrete as well.
• As a result, after the quantization step, we will
  deal with a discrete-time, finite-amplitude
  signal, in which each sample is represented by a
  finite number of bits.

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7.2.1 Scalar Quantization

• In scalar quantization
• Each sample is quantized into one of a finite
  number of levels which is then encoded into a
  binary representation.
• The quantization process is a rounding process
  each sampled signal point is rounded to the
  "nearest" value from a finite set of possible
  quantization levels.
• The set of real numbers R is partitioned into N
  disjoint subsets denoted by Rk,
• 1 ? k ? N (each called a quantization
  region).
• Corresponding to each subset Rk, a representation
  point (or quantization level) is chosen,
  which usually belongs to Rk.
• If the sampled signal at time i , xi belongs to
  Rk, then it is represented by , which is the
  quantized version of x.
• Then, is represented by a binary sequence
  and transmitted.
• ?This latter step is called encoding.
• Since there are N possibilities for the quantized
  levels, log2N bits are enough to encode these
  levels into binary sequences.
• Therefore, the number of bits required to
  transmit each source output is R log2 N bits.
• The price that we have paid for representing
  (rounding) every sample that falls in the region
  Rk by a single point is the introduction of
  distortion.

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Scalar Quantization

• Figure 7.3 shows an example of an 8-level
  quantization scheme.
• In this scheme, the eight regions are defined as
  R1 (-?, a1), R2 (a1, a2), ? , R8 (a8, -?).
• The representation point (or quantized value) in
  each region is denoted by and is shown in the
  figure.
• The quantization function Q is defined by

Figure 7.3 Example of an 8-level quantization
scheme.
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Scalar Quantization

• Depending on the measure of distortion employed,
  we can define the average distortion resulting
  from quantization.
• A popular measure of distortion, used widely in
  practice, is the squared error distortion defined
  as (x )2.
• In this expression x is the sampled signal value
  and is the quantized value, i.e.,
  Q (x).
• If we are using the squared error distortion
  measure, then
• where x - Q (x).
• Since X is a random variable, so are and
  therefore, the average (mean squared error)
  distortion is given by
• Mean squared distortion, or quantization noise as
  the measure of performance.
• A more meaningful measure of performance is a
  normalized version of the quantization noise, and
  it is normalized with respect to the power of the
  original signal.

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Uniform Quantization

• Uniform quantizers are the simplest examples of
  scalar quantizers.
• In a uniform quantizer, the entire real line is
  partitioned into N regions.
• All regions except R1 and RN are of equal length,
  which is denoted by ?.
• This means that for all 1 ? i ? N - 1, we have
  ail - ai ?.
• It is further assumed that the quantization
  levels are at a distance-of ?/2 from the
  boundaries a1, a2,..., aN-1 Figure 7.3 is an
  example of an 8-level uniform quantizer.
• In a uniform quantizer, the mean squared error
  distortion is given by
• Thus, D is a function of two design parameters,
  namely, a1 and ?.
• In order to design the optimal uniform quantizer,
  we have to differentiate D with respect to these
  variables and find the values that minimize D.
• Minimization of distortion is generally a tedious
  task and is done mainly by numerical techniques.
• Table 7.1 gives the optimal quantization level
  spacing for a zero-mean unit-variance Gaussian
  random variable
• The last column in the table gives the entropy
  after quantization.

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Uniform Quantization
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Nonuniform Quantization

• If we relax the condition that the quantization
  regions (except for the first and the last one)
  be of equal length, then we are minimizing the
  distortion with less constraints
• Therefore, the resulting quantizer will perform
  better than a uniform quantizer with the same
  number of levels.
• Let us assume that we are interested in designing
  the optimal mean squared error quantizer with N
  levels of quantization with no other constraint
  on the regions.
• The average distortion will be given by
• There exists a total of 2N - 1 variables in this
  expression (a1, a2, . . . , aN-1) and
  and the minimization of D is to be
  done with respect to these variables.
• Differentiating with respect to ai yields
• This result simply means that, in an optimal
  quantizer, the boundaries of the quantization
  regions are the midpoints of the quantized
  values.
• Because quantization is done on a minimum
  distance basis, each x value is quantized to the
  nearest

(7.2.10)
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Nonuniform Quantization

• To determine the quantized values , we
  differentiate D with respect to and define a0
  -? and aN ?.
• Thus, we obtain
• Equation (7.2.12) shows that in an optimal
  quantizer, the quantized value (or representation
  point) for a region should be chosen to be the
  centroid of that region.
• Equations (7.2.10) and (7.2.12) give the
  necessary conditions for a scalar quantizer to be
  optimal they are known as the Lloyd-Max
  conditions.
• The criteria for optimal quantization (the
  Lloyd-Max conditions) can then be summarized as
  follows
• 1. The boundaries of the quantization regions are
  the midpoints of the corresponding quantized
  values (nearest neighbor law).
• 2. The quantized values are the centroids of the
  quantization regions.

(7.2.12)
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Nonuniform Quantization

• Although these rules are very simple, they do not
  result in analytical solutions to the optimal
  quantizer design.
• The usual method of designing the optimal
  quantizer is to start with a set of quantization
  regions and then, using the second criterion, to
  find the quantized values.
• Then, we design new quantization regions for the
  new quantized values, and alternate between the
  two steps until the distortion does not change
  much from one step to the next.
• Based on this method, we can design the optimal
  quantizer for various source statistics.
• Table 7.2 shows the optimal nonuniform quantizers
  for various values of N for a zero-mean
  unit-variance Gaussian source.
• If, instead of this source, a general Gaussian
  source with mean m and variance ?2 is used, then
  the values of ai and read from Table 7.2 are
  replaced with m ?ai and m ? ,
  respectively, and the value of the distortion D
  will be replaced by ?2D.

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Nonuniform Quantization
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7.4.1 Pulse Code Modulation (PCM)

• Pulse code modulation is the simplest and oldest
  waveform coding scheme.
• A pulse code modulator consists of three basic
  sections a sampler, a quantizer and an encoder.
• A functional block diagram of a PCM system is
  shown in Figure 7.7.
• In PCM, we make the following assumptions
• The waveform (signal) is bandlimited with a
  maximum frequency of W. Therefore, it can be
  fully reconstructed from samples taken at a rate
  of fs 2W or higher.
• The signal is of finite amplitude. In other
  words, there exists a maximum amplitude xmax such
  that for all t , we have x(t) ? xmax.
• The quantization is done with a large number of
  quantization levels N, which is a power of 2 (N
  2v).

Figure 7.7 Block diagram of a PCM system.
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7.4.2 Differential Pulse Code Modulation (DPCM)

• PCM system
• After sampling the information signal, each
  sample is quantized independently using a scalar
  quantizer.
• Previous sample values have no effect on the
  quantization of the new samples.
• DPCM System
• When a bandlimited random process is sampled at
  the Nyquist rate or faster, the sampled values
  are usually correlated random variables.
• The exception is the case when the spectrum of
  the process is flat within its bandwidth.
• The previous samples give some information about
  the next sample
• This information can be employed to improve the
  performance of the PCM system.
• If the previous sample values were small, and
  there is a high probability that the next sample
  value will be small as well, then it is not
  necessary to quantize a wide range of values to
  achieve a good performance.

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Differential Pulse Code Modulation (DPCM)

• Figure 7.11 shows a block diagram of this simple
  DPCM scheme
• The input to the quantizer is not simply Xn
  Xn-1 but rather Xn
• We will see that is closely related to
  Xn-l, and this choice has an advantage because
  the accumulation of quantization noise is
  prevented
• The input to the quantizer Yn is quantized by a
  scalar quantizer (uniform or nonuniform) to
  produce
• Using the relations
  and
• At the receiving end, we have

Figure 7.11 A simple DPCM encoder and decoder.
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Differential Pulse Code Modulation (DPCM)

• ? Lena ???

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7.4.3 Delta Modulation

• Simplified version of the DPCM
• One bit quantizer with magnitudes with ??

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Delta Modulation

• A block diagram of a DM system is shown in Figure
  7.12.
• The same analysis that was applied to the simple
  DPCM system is valid
• Only one bit per sample is employed, so the
  quantization noise will be high unless the
  dynamic range of Yn is very low
• This, in turn, means that Xn and Xn-1 must have a
  very high correlation coefficient
• To have a high correlation between Xn and Xn-1,
  we have to sample at rates much higher than the
  Nyquist rate
• Therefore, in DM, the sampling rate is usually
  much higher than the Nyquist rate, but since the
  number of bits per sample is only one, the total
  number of bits per second required to transmit a
  waveform is lower than that of a PCM system

Figure 7.12 Delta modulation.
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Delta Modulation

• A major advantage of delta modulation is the very
  simple structure of the system.
• At the receiving end, we have the following
  relation for the reconstruction of
• Solving this equation for , and assuming
  zero initial conditions, we obtain
• This means that to obtain , we only have
  to accumulate the values of
• If the sampled values are represented by
  impulses, the accumulator will be a simple
  integrator
• This simplifies the block diagram of a DM system,
  as shown in Figure 7.13.

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Delta Modulation

• Step size ? Very important parameter in
  designing a delta modulator system
• Large values of ? cause the modulator to follow
  rapid changes in the input signal but at the
  same time, they cause excessive quantization
  noise when the input changes slowly.
• This case is shown in Figure 7.14 For large ?,
  when the input varies slowly, a large
  quantization noise occurs this is known as
  granular noise
• The case of a too small ? is shown in Figure 7.15
  In this case. we have a problem with rapid
  changes in the input.
• When the input changes rapidly (high-input
  slope), it takes a rather long time for the
  output to follow the input, and an excessive
  quantization noise is caused in this period.
• This type of distortion, which is caused by the
  high slope of the input waveform, is called slope
  overload distortion.

Figure 7.14 Large ? and Granular noise
Figure 7.15 Small ? and slope overload distortion
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Adaptive Delta Modulation

• We have seen that a step size that is too large
  causes granular noise, and a step size too small
  results in slope overload distortion
• This means that a good choice for ? is a "medium"
  value but in some cases, the performance of the
  best medium value (i.e., the one minimizing the
  mean squared distortion) is not satisfactory
• An approach that works well
• in these cases is to change the
• step size according to changes
• in the input
• If the input tends to change rapidly,
• the step size must be large so that
• the output can follow the input
• quickly and no slope overload
• distortion results
• When the input is more or less flat
• (slowly varying), the step size
• changed to a small value to prevent
• granular noise Figure 7.16.

Figure 7.16 Performance of adaptive delta
modulation.
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Transmission of Binary Data by RF Signals
Amplitude/Phase/Frequency Shift Keying
(ASK/PSK/FSK)
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Recommended Problems

• Textbook Problems from p369
• 7.1, 7.2, 7.6
• ??? ????? ??? ?? ???? ? ???? ?? ? PCM, DPCM, DM?
  ??? ???