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In this section, we discuss a more general

channel model called the band-limited coloured Gaussian Channel.

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The schematic diagram for this channel is the same

as the schematic diagram for the band-limited white Gaussian Channel.

The only difference is the noise process Z(t).

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Specifically, Z(t) is a zero mean additive coloured Gaussian noise,

meaning that the power spectral density is not necessarily a constant.

X prime of t and Z prime of

t, are filtered versions of X(t) and Z(t), respectively,

both band-limited to the interval [0,W].

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The output process, Y(t) is equal to X prime of t plus Z prime of t.

And Z prime of t is a band-limited coloured Gaussian noise, with the power

spectral density S_{Z prime} of f greater than and or equal to 0

for f from minus W to W and equal to zero otherwise.

As for the case of the white Gaussian channel, we regard X prime of t

as the channel input and Z prime of t as the additive noise process.

We also impose a power constraint P on the input process X prime of t.

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Recall that the capacity of the white Gaussian channel

band-limited to the interval [0,W], is equal to W times

log of 1 plus P over N_0 times W bits per unit time.

For the white Gaussian channel, band-limited to the interval [f_l,f_h],

where f_l is the lowest frequency and f_h is the highest frequency,

and f_l is a multiple of the bandwidth of the channel W equal to f_h minus f_l,

we can apply the bandpass version of the

sampling theorem to obtain the same capacity formula.

This model is called the bandpass white Gaussian Channel.

When f_l, the lowest frequency, is equal to 0, the

bandpass white Gaussian Channel reduces to the band-limited white Gaussian Channel.

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Here is the filter associated with the bandpass white Gaussian Channel.

Note that the lowest frequency f_l is a

multiple of W, the bandwidth of the channel.

And this is the schematic diagram for the bandpass white Gaussian Channel.

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Before we derive the capacity of the band-limited coloured Gaussian

Channel, let us first recall the assumptions of the channel model.

Z(t), the noise process, is a zero mean additive coloured Gaussian noise.

X prime of t and Z prime of t are filtered versions of X(t) and

Y(t) respectively, both band-limited to the interval [0,W].

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And the input power constraints on the input process X prime of t is equal to P.

This is an illustration of the power

spectral density of a band-limited coloured Gaussian Channel.

We now derive the capacity of such a channel.

First, divide the interval [0,W] into k subintervals.

The width of each subinterval is delta_k equal to W over k.

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Let the i-th subinterval be

[f_l^i,f_h^i] for i from one up to k.

As an approximation, assume that the

noise power over the i-th subinterval is a constant S_{Z,i}.

Then the channel consists of k sub-channels with

the i-th sub-channel being a bandpass white Gaussian

Channel occupying the frequency band [f_l^i,f_h^i].

Let P_i be the power allocated to the i-th sub-channel.

Then the capacity of the i-th sub-channel, is delta_k times

log of 1 plus P_i, divided by 2 times S_{Z,i}, times delta_k.

The workout of this expression is as follows.

For a white Gaussian Channel, which is band-limited to the integral [f_l, f_h],

where f_l is a multiple of the bandwidth W prime equals f_h minus f_l,

with noise level N_0 over 2, and power

constraint P, the capacity is W prime log

of one plus P divided by N_0 times W prime.

For the i-th sub-channel,

W prime is equal to delta_k,

P is equal to P_i,

and N_0 over 2 is equal to S_{Z,i}, or N_0 is equal to 2 times S_{Z,i}.

With these, we can obtain the expression in step five.

The noise process Z_i prime of t of the i-th sub-channel is obtained

by passing the noise process Z(t) through the corresponding

ideal bandpass filter band-limited to the interval [f_l^i,f_h^i].

It can be shown that the noise processes Z_i

prime of t, i from 1 up to k, are independent.

We will leave this as an exercise.

By sampling the sub-channels at a Nyquist rate 2 times delta_k,

where delta_k is the bandwidth of each sub-channel,

the k sub-channels can be regarded as a system of parallel Gaussian channels.

Thus the capacity of the channel is equal to the sum of the capacities

of the individual sub-channels when the

power allocation among the k sub-channels is optimal.

Let P_i star be the optimal power allocation for the i-th sub-channel.

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In step five, we obtained the capacity of the i-th sub-channel

when P_i is the power allocated to that sub-channel.

Then the capacity of the system of parallel Gaussian channels

is equal to summation i from 1 up to k, delta_k times log

of 1 plus P_i star, divided by 2 times S_{Z,i} times delta_k,

where the fraction inside the logarithm can be written

as P_i star over 2 times delta_k divided by S_{Z,i}.

And by proposition 11.23, the values of P_i star

over 2 times delta_k is equal to the positive part of nu

minus S_{Z,i}, where the water level nu can be obtained from the

constraint summation P_i star, i from one up to k, equals P.

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As k tends to infinity, where k is the number of subintervals, the capacity

of the system of parallel Gaussian channels given by this summation

tends to the integral from 0 to W log of 1 plus the

positive part of nu minus S_Z(f) divided by S_Z(f) df.

This can be written as one half times the same integral from minus W to W,

because S_Z(-f), is equal to S_Z(f).

That is the power spectral density is symmetrical about f equals 0.

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On the other hand, as k tends to infinity, summation i P_i star is equal

to summation i 2 times delta_k times the positive part of nu minus S_{Z,i}.

And this summation tends

to 2 times the integral

from 0 to W the positive

part of nu minus S_Z(f) df.

Again, this can be written as the integral from minus W

to W, the positive part of nu minus S_Z(f)df because the

power spectral density is symmetrical about f equals 0.

Therefore, the constraint summation i P_i star equals

P, tends to the constraint, the integral from minus

W to W, the positive part of nu minus S_Z(f)df equals P.

So the capacity of the band-limited coloured Gaussian Channel is given by

this formula, where the water level nu can be obtained by this constraint.

The optimal power allocation given in equation one

has a water-filling interpretation given in the next figure.

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Here, the water filling is for the range from minus W to W

with the total water volume equal to P.

And so the water volume on each side is equal to P over 2.