The Delta Modulator (DM)

The Delta Modulator is the first of two predictive coders that we’ll talk about in this book. Being a predictive coder, it works in just the way we saw in Figure 4.27. All we’ll do here is detail how this system creates its

x

_nP, and understanding that, we’ll explore in a bit of mathematical detail its inner workings.

How the DM creates an

xnP

This predictive coder creates the predicted value

x

_nP using a really simple idea: If you sample an incoming signal very quickly, then a good estimate of the current sample

x

n is simply the previous sample value

x

n−1. That is, a good predicted value of the sample

x

n is

x

n

P

=

x

n−1.

However, as one telecommunication engineer was quick to point out, there was a problem with this idea. “If you use

x

_nP=

x

_n−1,” she argued, “there’s no way to add back

x

_nP at the decoder (Figure 4.28), because

x

_n−1 is a sample of the input signal, and the decoder (in the receiver end of the channel) has no way to create exactly that.”

And with that the telecommunication engineers scratched their collective heads for a while, until one day one of them screamed, “I’ve got it! Have a look at this.” She pointed to the source decoder in Figure 4.28. “While the source decoder doesn’t have access to the

x

_n (or

x

_n−1), it does have access to

x

_n (and

x

_n−1). So, we can use

x

_n−1 as the predicted value, rather than

x

n−1.” That seemed a good idea, and everyone was in agreement. After some time and a bit of hard work, engineers decided to use

x

n−1 as the predicted value, that is,

x

_nP=

x

_n−1.

Some engineers decided to fine tune this idea, and as they played around with it, they found that a slightly better choice of

x

_nP is

x

_nP=a

x

_n−1, where a is a value very close to, but just a little less than, 1; specifically, they figured out that the optimal value of a was

a

R

R

x x

=

( )

( )

1

0

(4.53) where R_x(k) = E [x_n⋅ x_{n – k}].

The Block Diagram of the DM

Let’s now take a look at the block diagram of the DM to get an understanding of how this device is built. Both the DM and the source decoder are shown in the block diagram of Figure 4.29. The solid line shows that this DM and source decoder has the same form as the general predictive coder and source decoder of Figures 4.27 and 4.28. The dashed lines show the creation of the predictive value

x

n

P

=a

x

_n−1 at the DM and decoder.

Let’s take some time out now and describe how, in this block diagram, the value

x

_nP=a

x

_n−1 is created at the DM and decoder. We’ll start at the source decoder, because it’s easier. At the source decoder, the value

x

_nis available at the output. So, all we do— and the dashed lines of Figure 4.29 show this—is take the available

x

_n, delay it by one sample time (with the

z

−1 block) and multiply it by a; and then we’ve got it:

x

n

P

= a

x

_n−1. At the source coder, the value of

x

_n is not readily available, so what the dashed lines on the coder side of Figure 4.29 do is this: (1) first make

x

_n by adding

x

_nP and

E

_n (this is how

x

_n is made at the decoder), then (2) delay the

x

_n that’s just made and multiply it by a to get

x

n

P

=a

x

_n−1.

The Sampler and the Quantizer in the DM

While the block diagram of the DM is well-described by Figure 4.29, and a description of how it works is also given, there are a few things about the DM that have been left unsaid, and in this section I want to say them. These are things regarding the sampler and the quantizer.

The Sampler: First, we said earlier that the idea behind the DM was that if you sampled a signal fast enough, then the previous sample

x

_n−1 was a good predicted value for

x

_n, and we modified that slightly and came up with the predicted value of

x

_nP=a

x

_n−1, where a is a value close to 1. What I want to explore here is: how fast does the sampler in a DM actually sample the signal x(t) such that the sample

x

_n−1 is a very good predicted value of

x

_n? I don’t know exactly how telecommunication engineers came up with this, and I expect that it was trial and error, but ultimately when x(t) is a speech signal the sampler tends to work at four times the Nyquist rate, or eight times

Quantizer Symbol-to-bit Mapper + × + + × + x_n x_n xn E_n E_n E_n sampler ∧ ∧ ∧ ∧ bits ∧ xnp = axn–1 ∧ x_np _{= ax} n–1 Z–1 Z–1 a received bits Bit-to-symbol Mapper modulator, channel, demodulator ... + LPF + a

Delta modulator (DM) Source decoder_{for DM}

–

the Nyquist rate; that is, the sampler tends to work at eight times or sixteen times the maximum frequency of the incoming signal x(t).

The Quantizer: Next, let’s consider the quantizer. The quantizer maps the predicted error,

E

_n=

x

_n–

x

_nP, to an output

E

_n with one of N levels. How many output levels (N) does the quantizer use? The sampling rate in a DM is so high that the previous sample

x

_n−1 (and the value a

x

_n−1) is a very good predicted value for

x

_n, and so the error term

E

n=

x

n–

x

n

P

=

x

n–a

x

n−1 is very close to 0. Because of this, a quantizer can be built that is very good and very simple: the quantizer maps the input

E

n (close to 0) to an output with

one of only two levels, either +δ or –δ where δ is a value close to 0. The exact value of δ depends on the statistics of the input samples

x

_n.

In document McGraw Hill Telecommunications Demystified pdf (Page 120-122)