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This module introduces random signals.

Random signals are random variables which evolve, often with time (e.g. audio noise), but also with distance (e.g. intensityin an image of a random texture), or sometimes another parameter.

They can be described as usual by their cdf and either their pmf (if the amplitude is discrete, as in a digitized signal) ortheir pdf (if the amplitude is continuous, as in most analogue signals).

However a very important additional property is how rapidly a random signal fluctuates. Clearly a slowly varying signal suchas the waves in an ocean is very different from a rapidly varying signal such as vibrations in a vehicle. We will seelater in how to deal with these frequency dependent characteristics of randomness.

For the moment we shall assume that random signals are sampled at regular intervals and that each signal is equivalent to asequence of samples of a given random process, as in the following examples.

Detection of signals in noise: (a) the transmitted binary signal; (b) the binary signal after filtering with a half-sinereceiver filter; (c) the channel noise after filtering with the same filter; (d) the filtered signal plus noise at thedetector in the receiver.
The pdfs of the signal plus noise at the detector for the two ± 1 . The vertical dashed line is the detector threshold and the shaded area to the left of the origin represents theprobability of error when data = 1.

Example - detection of a binary signal in noise

We now consider the example of detecting a binary signal after it has passed through a channel which adds noise. Thetransmitted signal is typically as shown in (a) of .

In order to reduce the channel noise, the receiver will include a lowpass filter. The aim of the filter is to reducethe noise as much as possible without reducing the peak values of the signal significantly. A good filter for this has ahalf-sine impulse response of the form:

h t 2 T b t T b 0 t T b 0
Where T b = bit period.

This filter will convert the rectangular data bits into sinusoidally shaped pulses as shown in (b) of and it will also convert wide bandwidth channel noise into the form shown in (c) of . Bandlimited noise of this form will usually have an approximately Gaussian pdf.

Because this filter has an impulse response limited to just one bit period and has unit gain at zero frequency (the areaunder h t is unity), the signal values at the center of each bit period at the detector will still be ± 1 . If we choose to sample each bit at the detector at this optimal mid point, the pdfs of the signal plus noise atthe detector will be shown in .

Let the filtered data signal be D t and the filtered noise be U t , then the detector signal is

R t D t U t
If we assume that + 1 and -1 bits are equiprobable and the noise is a symmetric zero-mean process,the optimum detector threshold is clearly midway between these two states, i.e. at zero. The probability of error when thedata = + 1 is then given by:
D + 1 error D + 1 R t 0 F U -1 u -1 f U u
where F U and f U are the cdf and pdf of U . This is the shaded area in .

Similarly the probability of error when the data = -1 is then given by:

D -1 error D -1 R t 0 1 F U + 1 u 1 f U u
Hence the overall probability of error is:
error D + 1 error D + 1 D -1 error D -1 u -1 f U u D + 1 u 1 f U u D -1
since f U is symmetric about zero error u 1 f U u D + 1 D -1 u 1 f U u To be a little more general and to account for signal attenuation over the channel, we shall assume that the signalvalues at the detector are ± v 0 (rather than ± 1 ) and that the filtered noise at the detector has a zero-mean Gaussian pdf with variance σ 2 :
f U u 1 2 σ 2 u 2 2 σ 2
and so
error u v 0 f U u u v 0 σ f U σ u σ Q v 0 σ
where
Q x 1 2 u x u 2 2
This integral has no analytic solution, but a good approximation to it exists and is discussed in some detail in .

From we may obtain the probability of error in thebinary detector, which is often expressed as the bit error rate or BER . For example, if error 2 10 3 , this would often be expressed as a bit error rate of 2 10 3 , or alternatively as 1 error in 500 bits (on average).

The argument ( v 0 σ ) in is the signal-to-noise voltage ratio (SNR) atthe detector, and the BER rapidly diminishes with increasing SNR (see ).

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
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Can you compute that for me. Ty
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Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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progressive wave
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Random processes. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10204/1.3
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