Sampling is the process in which a continuous-time signal is sampled by
measuring its amplitude at discrete instants. In order to reconstruct
the signal correctly without any losses in data information. It is
necessary to sample the input signal at twice the signal frequency or
more. This criteria is called **Nyquist Sampling Frequency**. The
representation of this formula is shown below:

Quantization is the process of taking those discrete sampled values and representing them in step sizes. Assuming a message signal confined to the range (-mp, mp), this range is divided by L zones, each of step size (delta), given by:

A sample amplitude value is approximated by the midpoint of the interval in which it lies. An illustration of quantization is shown below.

The difference between the input and output signals of the quantizer becomes the quantization noise. This quantization error is thus represented with this formula below as the random variable between the two quantization intervals

Assuming that the error is equally likely to lie anywhere in the range (-(delta)/2, (delta)/2), the mean square quantizing error <qe**2> is given by :

Thus we finally have this equation after some derivation.

Encoding is described as the process of assigning a code number to the signal. For each step level produced by quantization, a numerical value is assigned. For instance, if a step level value is represented by the number seven as the seventh level; we can thus encode this level in a binary form such as 0111. This process as we see in end result is the process of taking an analog signal and digitizing it for digital transmission.

In ASK, the modulated signal can be expressed as

In PSK, the modulated signal can be expressed as

In FSK, the modulated signal can be expressed as

In order to determine which modulating scheme works best for our communications system, we must first determine the maximum noise tolerable for a constant bit rate of digital transmission. We will find the power spectral density of the noise as it relates to the probability of error in signal detection given these values.

**Signal Transmission Rate from source to submarine** - nFs <= Rb
(16)(48000) <= 768

Kbps = Rb

Input Voltage Signal - A = 1 V peak AC

(nu) - greek letter representing twice the power spectral density.

Period - T = 1/Rb = 1.302e-6 sec.

Probability of error - Perror = .00011

Using the above calculations we will calculate the maximum tolerable additive noise voltage where the signal can remain reliable at a constant rate of 768 Kbps.

For ASK modulating scheme:

A - Signal Power Pnoise = (nu)/2 - Power Spectral Density of noise signal

Vnoise = (Pnoise)(Signal Frequency)

Perror = Q(sqrt((A**2)T/4(nu) .0001 = Q(x) Looking at the table for the Q function we see that : x = 3.70

3.7 = sqrt((A**2)T/4(nu)) (nu) = ((1**2)(1.302e-6)/(4)(13.69)) (nu) = 2.38e-8

Pnoise = (nu)/2 = 1.189e-8

Vnoise = (Pnoise)(15.5Khz) = .184 mV

For ASK modulating scheme:

A - Signal Power Pnoise = (nu)/2 - Power spectral density of noise signal

Vnoise = (Pnoise)(Signal Frequency)

Perror = Q(sqrt((A**2)T/2(nu))) .00011 = Q(x) Looking at the table for the Q function wee see that: x = 3.70

3.7 = sqrt((A**2)T/(nu)) (nu) = ((1**2)(1.302e-6)/13.69) (nu)=9.51e-8

Pnoise = (nu)/2 = 4.76e-8

Vnoise = (Pnoise)(15.5 Khz) = .737 mV

We see that Phase Shift Keying is the best modulating scheme due to its ability to maintain a constant rate of transmission with a high degree of noise.

**Signal to Noise Ratio** - (S/N) = 1.76+6.02n = 1.76 + 6.02(16)
=98.08dB

where n - number of bits used in codec quantization.

We see that the signal tp noise ratio is greater than 96 dB which is good enough for CD-ROM quality.

Giventhis formula, will show the autocorrelation of a noisy waveform.
Since the autocorrelation is defined as the determining of similar
components; then a completely random function, like noise, will have an
autocorrelation approaching zero. In this **example**, the
autocorrelation and power of a noisy waveform are given.

r(t) = s(t) + n(t) where Rs(tau)=2exp(-tau) and Rn(tau) = exp(-2(tau))

Rr(tau) = E[r(t)t(t+(tau))]

Since the signal and noise are independent

E{s(t+(tau)n(t))} = E{s(t+(tau))}E{n(t)} = 0

and

E{s(t)n(t+(tau))} = E{s(t)}E{n(t+(tau))} = 0

Finally, the autocorrelation is:

Rr(tau)=Rs(tau)+Rn(tau)=2exp(-|(tau)|) + exp(-2|(tau)|)

The power spectral density isdefined as Gr=Rr(0) so Gf = 3 Watts.