Correlation Functions EEE3086F Signals and Systems II 2013

Correlation FunctionsA.J.Wilkinson, UCT EEE3086F Signals and Systems II212 March 7, 2013

EEE3086FSignals and Systems II

2013

A.J. Wilkinson

[email protected]

http://www.ee.uct.ac.zaDepartment of Electrical Engineering

University of Cape Town

Correlation FunctionsA.J.Wilkinson, UCT EEE3086F Signals and Systems II212 Page 2 March 7, 2013

2.7 Matched Filter

1. Goal of the Matched Filter2. Derivation of the Matched Filter 3. Examples

Contents:

Reference: F.G. Stremler, Introduction to Communication Systems, 3rd Edition


Goal of the Matched Filter

A.J.Wilkinson, UCT Correlation Functions EEE3086F Signals and Systems II212 Page 4 March 7, 2013

Aim and Applications of the Matched Filter

The matched filter is a linear filter that is designed to detect the presence of a waveform (or pulse) x(t) of known structure buried in additive noise. i.e. within a signal

where A and t0 are unknown constants, and n(t) is noise.

The output of a matched filter will typically exhibit a sharp peak in response to the presence of the desired waveform (or pulse) at its input. From this peak may be determined time location t

0, and amplitude A of the pulse within the received waveform.

E.g. 1. Radars transmit electromagnetic pulses which reflect off distance objects (or “targets”). The echoes are captured by a receiving antenna. The echoes may be very weak signals buried in additive noise (picked up by the antenna, and also created by the receiver itself). Matched filters are used to process the received waveform, and hence optimize the chance of detecting a target.

E.g. 2. In telecommunications systems using pulse code modulation, a bank of N filters may be constructed, with each filter specifically designed to respond to the presence of a particular pulse code. Comparison of the responses at the outputs identifies the received code.

v (t )=A x (t−t0)+ n(t)


Model of Transmitted and Received Signal with Additive Noise

Transmitted pulse

Received signal

where t0 is the propagation time delay,

A is an amplitude scaling factor, and n(t) is additive noise, described by its PSD S

n().

The SNR of the received pulse is defined as the ratio of the the peak signal power to average noise power, i.e.

SNR=peak signal power

average noise power=

max {∣A x (t−t0)∣2}

∣n(t)∣2=

∣A x (0)∣2

∣n(t)∣2

x (t)

v (t )=A x (t−t0)+ n(t)t 0

0


Definition of the Matched Filter

Given an input signalwhere x(t) is a pulse of known structure,t0 is an unknown delay, and A is an unknown scaling factor.

A “matched filter” is a linear filter h(t) or H() that is designed to optimize the peak SNR at its output at some specified instant relative to t

0.

The output is

The aim is to find h(t) or equivalently H() that maximizes(at some instant t

x, usually t

x >t

0):

v (t)=A x (t−t0)+ n(t)

peak SNR=∣y (t x)∣

2

∣no(t )∣2

vo(t)=v (t )⊛ h(t )= y (t )+ no (t )


Derivation of the Matched Filter


Simplified Signal Model

Let the input be:

Filter output:

Let

Output written as

vo(t )=v (t)⊛ h(t )=x (t )⊛ h(t )+ n(t )⊛ h(t )

v (t)=x (t )+ n(t )

vo(t )= y (t )+ no(t )

y (t )=x (t )⊛ h(t )

no(t )=n(t )⊛ h(t )

(output signal component)

(output noise component)

(where x(t) is a known waveform plus stationary noise n(t) )


Signal Model Illustrated

H()

noise

Inputsignal

Output

output SNR=∣y (t d)∣

2

∣no(t)∣2=

peak signal poweraverage noise power

td

t

t

peak

Maximize at t = td

vo(t )= y (t )+ no(t )

Filter

t

v (t)

n(t )

x (t)


Derivation of the Matched Filter

output SNR=∣y (t d)∣

2

∣no(t)∣2=

peak signal power at t=td

average noise power

y (t )=x (t )⊛ h(t )=1

2π ∫−∞

+ ∞

X (ω)H (ω)e jω t dω

∣y (t d)∣2=∣ 1

2π∫−∞

+ ∞

X (ω)H (ω)e jω td dω∣2

Peak outputsignal power

Average noisepower ∣no(t)∣

2=

12π ∫

−∞

+ ∞

S no(ω)dω =

12π ∫

−∞

+ ∞

S n(ω)∣H (ω)∣2dω

Output (signal component)

(from inverse FT)


Optimization Problem

The objective is to find a/the filter H() that optimizes the SNR at the time instant t = t

d:

∣y (td)∣2

∣no(t)∣2=

∣ 12π ∫

−∞

+∞

X (ω) H (ω)e jω t d dω∣2

12π

∫−∞

+∞


To do this, we make use of a (provable) mathematical relationship calledthe Schwartz (or Cauchy-Schwartz) inequality.


Schwartz Inequality for Integrals

The Schwartz inequality states that for two complex functions, f(x) and g(x), integrable over [a,b]

The equality holds (i.e. LHS = RHS) if where k is a real constant.

If then the left side is less than the right side.

∣∫a

b

f ( x )g (x)dx∣2

≤∫a

b

∣ f ( x)∣2dx ∫

a

b

∣g (x)∣2dx

g (x )=k f ∗( x)

g (x)≠k f ∗( x )

Proof? Google it!


To maximize SNR, apply Schwartz Inequality

∣y(td )∣2

∣no(t)∣2=

∣ 12π ∫

−∞

+ ∞

X (ω)H (ω)e jω t d dω∣2

12π ∫

−∞

+ ∞

S n(ω)∣H (ω)∣2 dω

∣y (t d )∣2

∣no(t)∣2

=∣ 12π

∫−∞

+ ∞

[ X (ω)

√S n(ω) ] [√S n(ω) H (ω)e jω t d ]dω∣2

12π ∫

−∞

+ ∞


∣y(t d )∣2

∣no(t)∣2

≤

12π ∫

−∞

+ ∞

∣ X (ω)

√S n(ω)∣2

dω1

2π ∫−∞

+ ∞

∣√S n(ω)H (ω)e jω td∣2dω

12π

∫−∞

+ ∞


f (x)≡[ X (ω)

√S n(ω) ] g (x )≡√S n(ω) H (ω)e jω t dApply Schwartz Inequality with:

Notice that the term on the right of the numerator cancels with the denominator.

SNR:

and

Rewrite expression as:


∣y (td )∣2

∣no(t )∣2≤

12π

∫−∞

+ ∞ ∣X (ω)∣2

S n(ω)dω

12π

∫−∞

+ ∞

S n(w)∣H (ω)∣2dω

12π

∫−∞

+ ∞


∣y (t d )∣2

∣n0(t )∣2

≤1

2π ∫−∞

+ ∞ ∣X (ω)∣2

S n(ω)dω

Swartz inequality becomes an “equality” i.e. LHS = RHS ifor in our case, if:

g (x )=k f ∗(x )

√S n(ω)H (ω)e jω td=k [ X (ω)

√S n(ω) ]∗

=> H (ω)=kX (ω)∗

S n(ω)e− jω td

The term on the rightof the numeratorcancels with the denominator.


Matched Filter

H (ω)=kX ∗

(ω)

S n(ω)e− jω t d

For the special case of white noise,

H (ω)=k X ∗(ω)e− jω td

h(t )=k x∗(−t+t d )

The filter that maximizesthe peak SNR at time t

d is:

k is a real constant, usually positive, and is often set to 1.

S n (ω)=constant

Inverse transforming gives:

Apply Fourier: X ∗(ω)→ x∗

(−t )

and Y (ω)e− jω t d→ y (t−t d)


Matched Filter (additive white noise)

H (ω)=X ∗(ω)e− jω t d

h(t)=x∗(−t+ t d)

t ωx (t )

tx (−t)

tt d

tt d

y (t )=x (t )⊛ h(t )≈c2 2BSa (2π B (t−t d))

ω

X (ω)

ω

Y (ω)=∣X (ω)∣2e− jω t d

−2π B 2π B

“Flip h(t), slide, mult. and integrate”

c

−2π B 2π B

≈c2 rect (ω/4π B)


Interpretation in Frequency Domain:

In the frequency domain, the matched filter magnitude boosts the frequency components for which the signal is large compared to the noise and suppresses those that are weak compared to the noise i.e.

The phase of the filter cancels the phase of X(), and hence the inverse Fourier transform integral gives a strong peak at t = t

d. In the white noise case

∣H (ω)∣∝∣X (ω)∣S n(ω)

H (ω)∝ X ∗(ω)

y (t )=1

2π ∫−∞

+ ∞

X (ω)H (ω)e jω t dω=12π ∫

−∞

+ ∞

∣X (ω)∣2e− jω td e jω t dω

H (ω)=kX ∗

(ω)

S n(ω)e− jω t d

H (ω)= X ∗(ω)e− jω t d

y (t d)=E x (by Parseval)


Interpretation in Time Domain

In the time domain, the matched filter for white noise is seen as the conjugate of a time-reversed (“flipped”) version of x(t).

It is easiest to consider the case of k=1 and td = 0, which is a

matched filter with “zero-delay”.

Performing a graphical convolution, “flips” h(t), which re-flips such that it will now match (or correlate well) with the “x(t)” contained in the received signal v(t) as one slides from left to right and integrates the product.

The “width” of the peak in the time domain depends on the bandwidth B of X(). To get y(t), we inverse transform:

h(t )=x∗(−t )

x∗(−t)

v (t)⊛ h(t )=x (t )⊛ h(t )+ n(t )⊛ h(t )

X (ω)H (ω)∝∣X (ω)∣2


Interpretation in Time Domain:

The width of the peak in the time domain depends on the bandwidth B of X(), since we are inverse transforming

If has an approximately rectangular bandwidth,then the output will have a Sa( ) shape.

The “3dB” width of the lobe will be [s].

∣X (ω)∣2

∣X (ω)∣2

δ t≈1

2 B

H()

noise

Bandlimitedinput signal

Output

td

tt

peak

Matched Filter

x (t )δ t≈

12 B


Physically Realizable Filters

The impulse response of the matched filter for waveform x(t) with additive white noise is

If this filter is to be constructed as a filter operating in “real-time”, then the output cannot respond before the input arrives.

Thus for “physical realizability”, the impulse response h(t) must begin after t = 0. i.e. h(t) = 0 for t <= 0.

In a real matched filter (built either with passive LRC components or using A/D, D/A and DSP technology), the MF peaks after the “entire waveform” has been fed into the input.

In designing a filter, one can always make it physically realizable by shifting h(t) to the right (by increasing t

d). This will simply delay the

output response by the corresponding amount.

h(t )=x∗(−t+ td )


Relationship to Correlation

Feeding a signal into a matched filtermatched to waveform (with white noise), is equivalent to a correlation operation between x(t) and v(t).

Recall correlation for energy signals:

The matched filter output is:

Rvx (τ )=∫−∞

∞

v*(t )x (t+ τ )dt

vo(t)=v (t)⊛ h(t )

vo(t) = ∫t '=−∞

∞

v (t ' )h(t−t ' )dt '=∫−∞

∞

v (t ' ) x∗(t '+ td−t )dt '

h(t )=x∗(−t+ t d)

vo(t ) = ∫u=−∞

∞

x∗(u)v (u+ [ t−t d ])du = Rxv (t−t d)

put u=t '+ t d−t

v (t)x (t)


Relationship to Correlation cont...

Thus if one correlates x(t) and v(t), and shifts the result to the right by an amount t

d, one gets the same output as from the

matched filter operation.

The matched filter output is

Note: if (the case where the input matches perfectly) then the MF output iswhich is just the autocorrelation function of x(t), shifted by t

d.

vo(t)=Rvx(t−t d)

vo(t)=v (t)⊛ h(t )

v (t)=x (t)vo(t)=Rx (t−t d)


Output SNR

The output SNR of the matched filter is

For white noise case,

Thus the output SNR is proportional to the energy of the received signal (and hence the energy of the transmitted signal).

∣y (td )∣2

∣n0(t)∣2

=12π ∫

−∞

+ ∞ ∣X (ω)∣S n(ω)

2

dω

(previously derived general case)

∣no(t )∣2

=12π ∫

−∞

+∞ η

2∣H (ω)∣

2 dω= =η

21

2π ∫−∞

+∞

∣X (ω)∣2 dω=

η

2E x

H (ω)=X ∗(ω)e− jω t d

⇒ ∣y ( t d)∣

2

∣n0(t)∣2

=E x

2

η

2E x

=E x

η/2

∣y (t d )∣2=∣ 1

2π ∫−∞

+∞

X (ω)H (ω)e jω t d dω∣2

=E x2

S n(ω)=η/2


Example of Matched Filter: detection of a received pulse

x( t )=A rect ( t−T /2T ) T

0

A

t

td+T

td

kA

t

Transmittedpulse.

Noisy receivedwaveform.

td

kA2T

Note: output peak is proportional to energy in pulse (A2T)

t

Delayed, scaled pulse.

Output of matched filter; SNR at peak is optimised.

Peak SNR

h(t )=x∗(−t )Matched Filter. T

A

t

h(t )⊛ v (t )

k x (t−t d)

v (t)=k x (t−t d)+ n(t )

td-Tt


Example of Matched Filter: detection of a received pulse

x (t )T0

A

t

td+T

td

kA

t

Transmittedpulse.

Noisy receivedwaveform.

tdNote: output peak is proportional to energy in pulse

t

Delayed, scaled pulse.

Output of matched filter; SNR at peak is optimised.

Peak SNR

h(t )=x∗(−t )Matched Filter.T

A

t

h(t )⊛ v (t )

k x (t−t d)

v (t)=k x (t−t d)+ n(t )

td-Tt

k Ex


Applications of Matched Filter

The matched filter is very useful for detecting known waveforms in the presence of additive noise .

E.g. 1. Radars use correlation receivers to detect weak echoes from distant targets. A replica of the transmitted pulse is correlated with the received waveform. The radar resolution (ability to separate close targets) depends on the bandwidth of the transmitted pulse.

E.g. 2. In telecommunications systems using pulse amplitude modulation, a matched filter, matched to the shape of the pulse, can be used to estimate optimally the pulse amplitude, since the height of the peak at the output of the matched filter is proportional to the amplitude of the received pulse.

h(t )⊛ v (t )

v (t)t

t

Received

Output of MF


EEE3086FSignals and Systems II

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Correlation Functions EEE3086F Signals and Systems II 2013

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