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**Narrowband Noise Representation**

In most communication systems, we are often dealing with band-pass filtering of signals. Wideband noise will be shaped into band limited noise. If the bandwidth of the band limited noise is relatively small compared to the carrier frequency, we refer to this as narrowband noise. We can derive the power spectral density G _{n}(f) and the auto-correlation function R_{nn}(Ï„) of the narrowband noise and use them to analyze the performance of linear systems. In practice, we often deal with mixing (multiplication), which is a non-linear operation, and the system analysis becomes difficult. In such a case, it is useful to express the narrowband noise as

(1)

where f_{c} is the carrier frequency within the band occupied by the noise. x(t) and y(t) are known as the quadrature components of the noise n(t). The Hibert transform of n(t) is

**Proof: **

The Fourier transform of n (t) is

Let (f) be the Fourier transform of . In the frequency domain, (f) = N(f)[-j sgn(f)]. We simply multiply all positive frequency components of N(f) by -j and all negative frequency components of N(f) by j. Thus

and the inverse Fourier transform of (f) is

= x(t) sin2Ï€f_{c} t + y(t) cos2Ï€f_{c}t

The quadrature components x(t) and y(t) can now be derived from equations (1) and (2).

x(t) = n(t)cos 2Ï€f_{c}t + sin 2Ï€f_{c}t (3)

and

y(t) = n(t)cos 2Ï€f_{c}t- sin 2Ï€f_{c}t (4)

Given n(t), the quadrature components x(t) and y(t) can be obtained by using the arrangement.

x(t) and y(t) have the following properties:

1. E[x(t) y(t)] = 0. x(t) and y(t) are uncorrelated with each other.

2. x(t) and y(t) have the same means and variances as n(t).

3. If n(t) is Gaussian, then x(t) and y(t) are also Gaussian.

4. x(t) and y(t) have identical power spectral densities, related to the power spectral density of n(t) by

G_{x}(f) = G_{y}(f) = G_{n}(f- f_{c}) + G_{n}(f+ f_{c} ) (5)

for fc - 0.5B < | f | < f_{c} + 0.5B and B is the bandwidth of n(t).

**Proof: **

Equation (5) is the key that will enable us to calculate the effect of noise on AM and FM systems. It implies that the power spectral density of x(t) and y(t) can be found by shifting the positive portion and negative portion of G_{n}(f) to zero frequency and adding to give G_{x} (f) and G_{y} (f).

In the special case where G_{n}(f) is symmetrical about the carrier frequency fc , the positive- and negative-frequency contributions are shifted to zero frequency and added to give

G_{x }(f) = G_{y}(f) = 2Gn(f- f_{c} ) = 2Gn(f+ f_{c} )

**Performance of Binary FSK: **

Consider the synchronous detector of binary FSK signals. In the presence of additive white Gaussian noise (AWGN), w(t), the received signal is

r(t) = Acos 2 f_{c1} t + w(t)

where A is a constant and fc1 is the carrier frequency employed if a 1 has been sent. The signals at the output of the band-pass filters of centre frequencies f_{c1 }and f_{c2} are

r_{1} (t) = Acos 2 f_{c1} t + n_{1} (t)

and

r_{2 }(t) = n_{2} (t)

where n_{1} (t) = x_{1} (t) cos 2 f_{c1} t - y1 (t) sin 2 f_{c1} t

and n_{2} (t) = x_{2} (t) cos 2 f_{c2} t - y_{2} (t) sin 2 f_{c2}t

are the narrowband noise.

With appropriate design of low-pass filter and sampling period, the sampled output signals are

vo_{1} = A + x_{1 }

vo_{2} = x_{2}

and

v = A + [x_{1} - x_{2} ].

x_{1} and x_{2} are statistically independent Gaussian random variables with zero mean and fixed variance Ïƒ^{2 }= N, where N is the power of the random variable. It can be seen that one of the detectors has signal plus noise, the other detector has noise only.

When f_{c2 }is the carrier frequency employed for sending a 0, the received signal is r(t) = Acos 2 f_{c2} t + w(t). It can be

shown that

v = -A + [x_{1} - x_{2} ]

Since E [x_{1} - x_{2} ]^{2} = E [x_{ 1} ]^{2} - 2E [x _{1} x_{2} ]^{2} + E [x_{2 }]^{2} = E [x_{1} ]^{2} + E [x_{2}]^{2} = 2 + 2, the total

variance

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