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Random Process: Some Examples

• A sinusoid with random phase

Consider a sinusoidal signal with random phase, defined by

X(t) = a sin(ω0t + Θ)

where ω0 and a are constants, and Θ is a random variable that

is uniformly distributed over a range of 0 to 2π (see Figure 1)

fΘ(θ) =1

2π, 0 ≤ θ ≤ 2π

= 0, elsewhere

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Θ

ω 0sin( t+Θ)

Figure 1: A sinusoid with random phase

This means that the random variable Θ is equally likely to

have any value in the range 0 to 2π. The autocorrelation

function of X(t) is

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RX(t) = E[X(t + τ)X(t)]

= E[sin(ω0t + ω0τ) + Θ) sin(ω0t + Θ)]

=1

2E[sin(2ω0t + ω0τ + 2Θ)] +

1

2E[sin(ω0τ)]

=1

2

∫ 2π

0

1

2πcos(4πfct + ω0τ + 2Θ)] − cos(ω0τ) dΘ

The first term intergrates to zero, and so we get

RX(τ) =1

2cos(ω0τ)

The autocorrelation function is plotted in Figure 2.

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ω 0cos( 1 τ)

2πω 0

Figure 2: Autocorrelation function of a sinusoid with random phase

The autocorrelation function of a sinusoidal wave with random

phase is another sinusoid at the same frequency in the τ -

domain

• Random binary wave

Figure 3 shows the sample function x(t) of a random process

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X(t) consisting of a random sequence of binary symbols 1 and

0.

+1

−1

t delay

Figure 3: A random binary wave

1. The symbols 1 and 0 are represented by pulses of amplitude

+1 and −1 volts, respectively and duration T seconds.

2. The starting time of the first pulse, tdelay, is equally likely

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to lie anywhere between zero and T seconds

3. tdelay is the sample value of a uniformly distributed random

variable Tdelay with a probability density function

fTdelay(tdelay) =

1

T, 0 ≤ tdelay ≤ T

= 0, elsewhere

4. In any time interval (n − 1)T < t − tdelay < nT , where n is

an interger, a 1 or a 0 is determined randomly (for example

by tossing a coin: heads =⇒ 1, tails =⇒ 0

– E[X(t)] = 0, for all t since 1 and 0 are equally likely.

– Autocorrelation function RX(tk, tl) is given by

E[X(tk)X(tl)], where X(tk) and X(tl) are random variables

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Case 1: when |tk − tl| > T . X(tk) and X(tl) occur in different

pulse intervals and are therefore independent:

E[X(tk)X(tl)] = E[X(tk)]E[X(tl)] = 0, for|tk − tl| > T

Case 2: when |tk − tl| < T , with tk = 0 and tl < tk. X(tk) and

X(tl) occur in the same pulse interval provided tdelay satisfies

the condition tdelay < T − |tk − tl|.

E[X(tk)X(tl)|tdelay] = 1, tdelay < T − |tk − tl|

= 0, elsewhere

Averaging this result over all possible values of tdelay, we get

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E[X(tk)X(tl)] =

∫ T−|tk−tl|

0

fTdelay(tdelay) dtdelay

=

∫ T−|tk−tl|

0

1

Tdtdelay

= (1 −|tk − tl|

T), |tk − tl| < T

The autocorrelation function is given by

RX(τ) = (1 −|τ |

T), |τ | < T

= 0, |τ | > T

This result is shown in Figure 4

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1

T−T

Figure 4: Autocorrelation of a random binary wave