This phenomenon is referred to as the filter roll-off generally expressed in Decibels of attenuation per octave of frequency. When a filter is designed there is the intention of making the roll-off as narrow as achievable which gives the filter a chance to get as close as possible to the intended design in terms of performance. There is a difference that exists between the upper and the lower cutoff frequencies which is referred to as the bandwidth of the filter while the ratio of bandwidths which is obtained by using two distinct attenuation values in order to find the cutoff frequency is referred to as the shape factor. For instance, when the shape factor is said to be 2:1 at 30/3 dB then it means that the bandwidth obtained between frequencies at 30 dB attenuation is double that obtained at 3 dB attenuation. The electrical symbol of a band pass filter is as shown below (schematic).

According to Hasan (1991), the extensive test of the Phase Locked Loop (PLL) FM demodulator in Gaussian modulation is replicated in consideration of additive noise and FM interference by means of the Monte Carlo method. The modulating Gaussian random signals are simulated by sums of sine waves of equally spaced frequencies and random phases.

Monte Carlo simulation

By the Monte Carlo method, the Gaussian message ?s (t) of bandwidth Ws rad/s and rms frequency deviation ?s rad/s is simulated by a sum of Ns sine waves

(1)

Where is the peak frequency deviation of the nth tone, is the fundamental modulation frequency so that Ns

a = Ws, and is a random phase distributed uniformly over (-

). Analogously, the Gaussian message ?i (t) is simulated by (2)

Where, and Ni

a = Wi are, respectively, the peak frequency deviation of the nth tone, the rms frequency deviation and the bandwidth of ?i (t), and is a random phase distributed uniformly over (-

). If the numbers Ns and Ni of tones simulating the Gaussian messages are large enough, the statistics of (1) and (2) approach that of Gaussian noise.

Omitting the...

The amplitude modulation and the phase modulation of the filtered FM signals are calculated with arbitrary accuracy depending on the number of spectral components considered. The distorted frequency modulation ?s (t) of the filtered desired signal is obtained analytically as the derivative of ?s (t) with respect to t. The colored Gaussian noise processes nc (t) and ns (t) are simulated using the FFT-based filtering as follows. Uniform deviates are generated by a routine based on three linear congruential generators. Normal deviates are obtained from uniform deviates using the Box-Muller method and the resultant Gaussian processes are filtered in the frequency domain by the transfer function HLP (i
A fast recurrent algorithm

3

With linear prediction

4

Computational efficiency of the Monte Carlo simulation is enhanced by implementing the time-consuming Fourier method using a low sampling rate. The fast recurrent algorithm based on the above equations (3) -- (4) allows a high sampling rate with the amplitude and phase modulation of both filtered FM signals as well as the distorted frequency modulation of the desired signal cubic spline interpolated. (Hasan, 1991, p. 2-3).