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Guassian Random Process and White Noise

Lecture 15 EEE 352 Analog Communication Systems Mansoor Khan Electrical Engineering Dept. CIIT Islamabad Campus

Gaussian Processes • Gaussian process is a special random process – Important for communications because noise is usually modeled as Gaussian process

• Definition – A random process X(t) is Gaussian process if at any time instant ti, the random variable X(ti) has Gaussian distribution (is Gaussian random variable)

• A simple property – when a Gaussian process passes a linear filter, the output is still a Gaussian process

X(t)

X(ti) ti

t

• Gaussian process & Gaussian random variable – Remind: if X(t) is Gaussian process, then at each time instant ti, X(ti) is Gaussian random variable – Gaussian random variable X(ti) described by a PDF with mean μ and variance σ2

• Gaussian process & Gaussian random variable – Remind: if X(t) is Gaussian process, then at each time instant ti, X(ti) is Gaussian random variable – Gaussian random variable X(ti) described by a PDF with mean μ and variance σ2

– Gaussian random process X(t) described by mean function μ(t) and power spectral density (PSD) SX(f) • or use autocorrelation function RX(τ) in place of PSD

– Main property #1: for stationary random process

mean function of random process is contant, and equals to mean of random variable, (t ) – Main property #2: power of random process equals to power of random variable

S X ( f )df E X (ti ) X (ti )

– Main property #3: If random process has zero mean, then

S X ( f )df

2

• Example – Let X(t) be zero-mean stationary Gaussian process with PSD SX(f). Determine PDF of X(3).

5, for 500 f 500 Hz SX ( f ) else 0, Mean of random variable X (3) : X X (t ) E[ X (t )] 0 variance of random variable X (3) :

500

500

S X ( f )df 2

5df 5000

Probability density function of random variable X (3) : 1 f ( x) e 10000

x2 10000

White Processes • White process is a special random process with constant power spectral density – Power density is a constant for all frequencies

SW ( f ) C, RW ( ) C ( ) – White process is an ideal math model because the total power is infinite, analogy to impulse function δ(t). – But it is useful to model noise, and usually gives very good approximation and extremely simple math.

Noise • An important type of noise is thermal noise – The origin is random movement of electrons – This noise signal can be modeled as Gaussian process • The composite effect of a large number of small electrons becomes Gaussian distributed

– We also model the noise signal by white process • Without any special filtering, noise power will be almost constant for all frequencies

– The noise signal is usually added to information signals, so we call it Additive White Gaussian Noise (AWGN)

• Model of AWGN (additive white Gaussian noise)

X(t): random or deterministic signal W(t): AWGN Y(t)=X(t)+W(t): random process

A realization of AWGN W(t)

• AWGN W(t) is a white Gaussian random process – “White”: constant PSD

N0 SW ( f ) , 2

N0 RW ( ) ( ) 2

– “Gaussian”: at time ti, random variable W(ti) has Gaussian distribution, usually assumed with zero mean.

x2

2 1 f ( x) e 2 2

Filtered Noise Processes • AWGN is an ideal noise model, we often use it to describe the received noise – Practical noise can usually be modeled by filtering AWGN, i.e., processing noise via filters

• Major difference between AWGN and filtered noise – PSD different: filtered noise has limited bandwidth B, and PSD: . – Distribution of random variable same: still Gaussian random variable

SN ( f ) | H ( f ) |2 SW ( f )

• Example – If AWGN pass an ideal lowpass filter with bandwidth B and magnitude 1, what is the output signal N(t)? Determine the PDF of N(0).

N (t ) is still Gaussian process, with PSD 2 2 N0 S N ( f ) | H ( f ) | SW ( f ) | H ( f ) | 2 N0 , for B f B 2 0, otherwise

1, for B f B H( f ) otherwise 0,

N (0) is Gaussian random variable with zero-mean, variance B N 0 N2 S N ( f )df df N 0 B B 2 Its PDF is: f ( x)

1 e 2 N 0 B

x2 2 N0 B

• Random process & random variable – When an AWGN signal passes a filter with bandwidth B and unit passband magnitude • If AWGN has zero mean, then all the random processes and random •

variables are zero mean. PSD and variance relationship

Random process PSD S(f) Input W(t)

Output N(t)

Random variable variance σ2

N0 SW ( f ) 2 N0 , for f in passband SN ( f ) 2 0, otherwise

N0 B 2 N

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Lecture 15 EEE 352 Analog Communication Systems Mansoor Khan Electrical Engineering Dept. CIIT Islamabad Campus

Gaussian Processes • Gaussian process is a special random process – Important for communications because noise is usually modeled as Gaussian process

• Definition – A random process X(t) is Gaussian process if at any time instant ti, the random variable X(ti) has Gaussian distribution (is Gaussian random variable)

• A simple property – when a Gaussian process passes a linear filter, the output is still a Gaussian process

X(t)

X(ti) ti

t

• Gaussian process & Gaussian random variable – Remind: if X(t) is Gaussian process, then at each time instant ti, X(ti) is Gaussian random variable – Gaussian random variable X(ti) described by a PDF with mean μ and variance σ2

• Gaussian process & Gaussian random variable – Remind: if X(t) is Gaussian process, then at each time instant ti, X(ti) is Gaussian random variable – Gaussian random variable X(ti) described by a PDF with mean μ and variance σ2

– Gaussian random process X(t) described by mean function μ(t) and power spectral density (PSD) SX(f) • or use autocorrelation function RX(τ) in place of PSD

– Main property #1: for stationary random process

mean function of random process is contant, and equals to mean of random variable, (t ) – Main property #2: power of random process equals to power of random variable

S X ( f )df E X (ti ) X (ti )

– Main property #3: If random process has zero mean, then

S X ( f )df

2

• Example – Let X(t) be zero-mean stationary Gaussian process with PSD SX(f). Determine PDF of X(3).

5, for 500 f 500 Hz SX ( f ) else 0, Mean of random variable X (3) : X X (t ) E[ X (t )] 0 variance of random variable X (3) :

500

500

S X ( f )df 2

5df 5000

Probability density function of random variable X (3) : 1 f ( x) e 10000

x2 10000

White Processes • White process is a special random process with constant power spectral density – Power density is a constant for all frequencies

SW ( f ) C, RW ( ) C ( ) – White process is an ideal math model because the total power is infinite, analogy to impulse function δ(t). – But it is useful to model noise, and usually gives very good approximation and extremely simple math.

Noise • An important type of noise is thermal noise – The origin is random movement of electrons – This noise signal can be modeled as Gaussian process • The composite effect of a large number of small electrons becomes Gaussian distributed

– We also model the noise signal by white process • Without any special filtering, noise power will be almost constant for all frequencies

– The noise signal is usually added to information signals, so we call it Additive White Gaussian Noise (AWGN)

• Model of AWGN (additive white Gaussian noise)

X(t): random or deterministic signal W(t): AWGN Y(t)=X(t)+W(t): random process

A realization of AWGN W(t)

• AWGN W(t) is a white Gaussian random process – “White”: constant PSD

N0 SW ( f ) , 2

N0 RW ( ) ( ) 2

– “Gaussian”: at time ti, random variable W(ti) has Gaussian distribution, usually assumed with zero mean.

x2

2 1 f ( x) e 2 2

Filtered Noise Processes • AWGN is an ideal noise model, we often use it to describe the received noise – Practical noise can usually be modeled by filtering AWGN, i.e., processing noise via filters

• Major difference between AWGN and filtered noise – PSD different: filtered noise has limited bandwidth B, and PSD: . – Distribution of random variable same: still Gaussian random variable

SN ( f ) | H ( f ) |2 SW ( f )

• Example – If AWGN pass an ideal lowpass filter with bandwidth B and magnitude 1, what is the output signal N(t)? Determine the PDF of N(0).

N (t ) is still Gaussian process, with PSD 2 2 N0 S N ( f ) | H ( f ) | SW ( f ) | H ( f ) | 2 N0 , for B f B 2 0, otherwise

1, for B f B H( f ) otherwise 0,

N (0) is Gaussian random variable with zero-mean, variance B N 0 N2 S N ( f )df df N 0 B B 2 Its PDF is: f ( x)

1 e 2 N 0 B

x2 2 N0 B

• Random process & random variable – When an AWGN signal passes a filter with bandwidth B and unit passband magnitude • If AWGN has zero mean, then all the random processes and random •

variables are zero mean. PSD and variance relationship

Random process PSD S(f) Input W(t)

Output N(t)

Random variable variance σ2

N0 SW ( f ) 2 N0 , for f in passband SN ( f ) 2 0, otherwise

N0 B 2 N

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