Download Optimum Noise Filtering

Optimum Noise Filtering
Overview of Noise Filtering
 Introduction: The goal of noise filtering is to extract the signal of interest from
random noise.
 Important factors:
o Signal information
o Noise information
o Signal-to-noise (S/N) ratio
o Distortion
o Implementation
 Signal information:
o Deterministic signals–The signal of interest is known. The goal is to detect the
signal with the presence of noise. For examples: Radar or sonar echoes,
tracking device signals, and digital signals
o Random signals–Only the statistical properties of the signal of interest are
known. The goal is to detect and/or estimate the signal with the presence of
noise. For examples: Radio signals, image data, and seismic signals.
 Noise information:
o White noise – The spectrum of white noise is constant. This assumption is
appropriate for many kinds of noise phenomena. White noise can be handled
easily but sometimes this assumption oversimplifies the problem.
White noise

o Band-limited white noise – The spectrum of band-limited white noise is
constant within a specified frequency band and is zero outside the band.
Unlike a white noise process, a band-limited white noise process has finite
energy, a requirement that must be satisfied for any physical process.
Low-pass
white noise

Band-pass
white noise

o 1/f noise – The spectrum of 1/f noise is proportional to 1/f . As a result, the
1/f noise process mainly produces noise components at low frequencies.
1/f noise

o Other noise types – Man-make noise, atmospheric disturbances, and …



Signal-to-noise (S/N) ratio:
o Maximizing signal energy – To maximize received signal energy we need to
increase the gain (decrease the loss) at the frequency bands where the signal
of interest is strong compare to noise.
o Minimizing noise energy – To minimize received noise energy we need to
decrease the gain (increase the loss) at the frequency bands where the signal
of interest is weak compare to noise. For example, the gain should be 0 (loss =
) at frequencies where the signal energy is negligible.
o The gain should be zero (the loss should be infinite) outside the signal
frequency band.
Distortion:
o Maximum S/N ratio usually requires the system to have a non-uniform
frequency response.
o Non-uniform frequency responses introduce distortion because signal
components at different frequencies are amplified (attenuated) differently.
o A practical system has to consider the trade-offs between noise and distortion.
Implementation:
o A physical system cannot provide infinite energy – an all-pass filter is
impossible to build.
o A physical system must be casual – discontinuities in the response spectrum
cannot be realized.
o A physical system must be stable – RHP poles are not allowed.
Matched Filters
 Introduction: When the signal of interest is known and the goal is to detect its
presence, a matched filter is used to maximize the S/N ratio of the detection process.
 Basic principles:
o Background noise is assumed to be white noise or band-limited white noise
with the noise bandwidth encompasses the entire signal spectrum.
o The signal of interest is deterministic.
o Background noise is assumed to be additive: Input = signal + noise
s(t)

n(t)
y(t)
h(t)
yo(t)

o If the frequency response of the filter matches the frequency spectrum of the
signal, the S/N ratio is maximized.
 At frequencies where the signal is strong, the system gain should also
be high to enhance the difference between the signal and the noise.
 At frequencies where the signal is weak, the system gain should also
be low to de-emphasize the noise energy at these frequencies.
 At frequencies where the signal energy is negligible, the system gain
should be zero to filter out the out-of-band noise.
o Because of the non-uniform frequency response, the filter output is distorted.
o This technique is ideal for signal detection since in this case only the signal
energy (with respect to noise), not the actual shape, is.
o This technique is not useful if the shape of the signal waveform is unknown.
Mathematical details:
o From linear system theory the output is obtained from the convolution
integral:


yo (t )   h(t   ) y ( )d   h( ) y (t   )d






  h( ) s (t   )d   h( )n(t   )d  so (t )  no (t )
For a white noise input with unity magnitude the S/N ratio at t = T is:
2
(S / N ) 2
T


 2
  2

h
(

)
s
(
T


)
d

h
(

)
d




    s (T   )d 
2
s (T )
   
  

 o 2
   

E[no (T )]
2
2
 h ( )d
 h ( )d


Note: The inequality is the direct result of the Schwarz inequality. The
equality holds if the following is true:

h

2
( )d 

s
2
(T   )d , which means: ( S / N )


2
T,
max
  s 2 (T   )d .

o In general, the following filter response can maximize the S/N ratio if the
autocorrelation function (or the auto-spectrum) of the input noise is known:
 S ( f )   j 2 f T
H( f )  
where  is an arbitary constant, T is the
e
 S nn ( f ) 
observatio n period, and S nn ( f ) is the autospectr um of the noise process.
o For white noise process, the impulse response of the matched filter is:
h(t )  s (T  t ) for t  0
s(t)
0
h(t)
T
t

Numerical examples:
o Rectangular pulses:
Noise
Noise Spectrum
5
N
n
20 log
0
5
5
S
20
Signal
100
n
0
20
200
1
n
20 log
S_F
n
5
Signal Spectrum
10
100
n
0
0
n
N_F
20
40
100
Signal + Noise
n
200
1
10
n
100
Frequency10Response
100
S + N Spectrum
5
0
T
20 log
0
n
T_F
n
20
5
5
h0
Impulse 100
Response
200
n
0
20 log
0
n
1
n
H0
n
5
20
40
100
n
200
1
Output
10
n
100
Output Spectrum
40
400
out
2
n
20 log
200
0
out_F
n
20
0
100
n 20
200
1
10
n
100
o Saw-tooth pulses:
Noise Spectrum
Noise
5
0 if ( n  W )
Shapen
n
20
W
(n W
otherwise
20
N
n
20 log
0
N_F
n
5
20
100
Signal n
200
1
100
0
20 log
0
n
10
Signal nSpectrum
5
S
0
S_F
n
5
Tn
Sn
Nn
20
40
100
Signal + Noise
n
200
1
10
100
S + Nn Spectrum
5
0
T
20 log
0
n
T_F
hm  n
0 if ( n  m)
n
n
m
otherwise
20
20
5
100
n
Impulse Response
200
1
5
h0
20 log
H0
n
5
200
1
2
10
100
Output nSpectrum
100
n
20
40
100
Output n
out
100
0
0
n
10
n Response
Frequency
40
20 log
50
out_F
n
20
0
0
100
n 20
200
1
10
n
100
h0n
h 900  1000
(n m
n
2

Examples:
o White noise process:
s(t)  e -t for t  0 
-j 2 f T
e
H( f ) 
1  j 2 f
S( f ) 

 h(t )  
0
o Realistic low-pass noise:
y (t )  Acos 2 f C t  n(t ) for

s (t )

1
1  j 2 f
e

-j 2 f (T  t )
1  j 2 f
 -(T  t ) for t  T
df  e
0
otherwise
0t T
A
 ( f  f C )   ( f  f C )
2

E[n(t )n(t   )]  e
S ( f )  F{ Acos 2 f c t} 
E[n(t )]  0,

 S * ( f ) j 2 f T 
 S nn ( f )  F{e
}
 hF 
e

1  4 2 f 2
 S nn ( f )

A
A
e  2 f (T  t )  e 2 f (T  t )  

cos 2 f C (T  t )
2 
2

 1  4 2 f C 2
1  4 f C 

constant

2
1
Radar echo detection:
X
Matched
filter
Detector

Display
Matched filter implementation
Matched filter
y(t)


T
0
yo(t)
s(t)
T
T
T
0
0
0
yo (t )   y( ) h(T   ) d   y( ) s(T  T   ) d   y( ) s( ) d
Wiener Filters
 Introduction: The matched filter technique requires the exact knowledge of the signal
and thus is used for detection only. If we need to reconstruct a signal with minimum
distortion and only the statistical properties of the signal is available, the Wiener filter
technique can be used for this purpose (signal estimation).
 Basic principles:
o The filter that minimizes the mean-square error (minimum distortion) has the
following transfer function H( f ).
S (f)
S ss ( f )
H ( f )  ss

S yy ( f ) S ss ( f )  S nn ( f )
o It is called the Wiener filter. More information on optimum filtering can be
found in Cooper; Probabilistic Methods of Signal and System Analysis, 2nd
edition, p. 402-411.
o The above transfer function can be implemented numerically for pre-recorded
data but is not physically realizable because it is non-casual. The following
steps convert the non-casual Wiener filter into a casual one.
1) Express Syy( f ) as a product of two factors:
S yy ( f )  S ss ( f )  S nn ( f )  Ayy ( f ) Ayy ( f )
where Ayy ( f ) has all of the LHP poles and zeros and Ayy ( f ) has all of
the RHP poles and zeros. Split the poles and zeros on the j axis
evenly to both planes.
j
2) Use partial fraction expansion:
S ys ( f )
 B ( f )  B ( f )

Ayy ( f )
where B  ( f ) has all of the LHP poles and zeros and B  ( f ) has all of
the RHP poles and zeros. Split the poles and zeros on the j axis
evenly to both planes.
Note: if n(t) and s(t) are uncorrelated, S ys  f   S ss  f 
3) The transfer function of the casual Wiener filter is given by:

B ( f )
B  ( f ) j 2 f t
H( f )  
h(t )   
e
df
Ayy ( f )
A (f)
 yy

Example:
t
1) Rss (t )  e
S nn  N 0
t
S ss ( f )  F{Rss (t )}  F{2e } 
2
 S ys ( f )
1  4 2 f 2
For a non-casual filter:
S ss ( f )
2 1  4 2 f 2
1
H( f ) 


2 2
S ss ( f )  S nn ( f ) 2 1  4 f  N 0 1  N 0 2  1  (2 f ) 2
 
 


S yy
S yy
S yy
Step 1:
N 0 2  N 0  N 0  (2 f ) 2
2
S yy  S ss ( f )  S nn ( f ) 

N

0
1  4 2 f 2
1  (2 f ) 2







N0

2  N 0 

2  N 0 
N 0  j 2 f 
(1  j 2 f )(1  j 2 f )
Step 2:
S ys ( f )
A (f)


N0
N0


N0

2  N 0 
S ys ( f )
Ayy ( f )


2

N 0  j 2 f (1  j 2 f )

N0

2  N 0 
N 0  j 2 f

1  j 2 f
 1  j 2 f 
j 2 f 
 2 N 0 

N0


1
N0 1 

 1  j 2 f


2  N 0 

N0
N0






,
1
2  N 0  N 0  j 2 f


N0 1
1  j 2 f

N0 1
K2
2  N 0  N 0  j 2 f
1
 K1
1  2  N 0  N 0

B ( f )

Ayy ( f )
 B
 K1


 1  j 2 f


2
1



N 0 1  2  N 0  N 0  1  j 2 f
Step 3:
H0 ( f ) 
2  N 0 
j 2 f  1
N0
2
N0
1


2  N 0 
 B ( f ) 
h0 (t ) 
Ayy ( f ) 
,
2 1  4 2 f 2
2  N 0  N 0  j 2 f






2 (1  j 2 f )(1  j 2 f )
2  N 0  N 0  j 2 f 1  j 2 f 
1
1  j 2 f
 N0
N 0  j 2 f

N 0  j 2 f
1  j 2 f
1
2  N 0  N 0  j 2 f
K1 
K2 
2  N 0 
S ss ( f )

Ayy ( f )


yy

N0
 Ayy ( f ) 

1
2  N 0  N 0  j 2 f
,
B ( f ) 




2  N 0  N 0  1
2  N 0  N 0  j 2 f
N0






2  N 0  N 0  1 1  j 2 f 
2  N 0  N 0  1

2  N 0  N 0  j 2 f  1  j 2 f  2  N 0  N 0  j 2 f
( f ) Ayy ( f ) e j 2 f t df 

2  N 0 
 
N 0  1 exp 
2  N 0 

N 0 t  u (t )
Appendix: Noise Suppression (Paul Packebush; How to keep instruments accurate inside
hot, noisy PCs; EDN, 4-13, 2000, p. 189)
 Introduction:
o PCs are less expensive and more powerful now.
o They are being used in more and more measurement and control systems.
o Often, the instruments are actually inside the PC.
o Within a PC’s hot, noisy environment accurate measurements of low voltage
and high-frequency signals are difficult to make.
o Both data-acquisition cards as well as digital components present problems.
o Instrumentation designers usually have no control – components, chassis,
power supplies, and other plug-in peripherals create an unpredictable
environment.
 Identifying the major sources of troubles:
o Temperature variation: 5 to 15C higher than the ambient temperature.
 Power supply dissipation
 Reduced airflow
 Hot CPU & Peripherals
o EM fields: EM emission from other devices
 Radiated EMI: Propagated through space
 Conducted EMI: Conducted through ground or power buses
 Video systems and power supplies are the worst.
 Video systems: Video cards, cables, and monitors.
 Monitors: Radiated EMI from the fly-back dc/dc converters presents
both high- and low-frequency interference–Horizontal scan (15.74
KHz to 80 KHz) and vertical scan (40 to 80 Hz).
o Power buses management (ground): Small radiated EMI but large conducted
EMI–Switching noise to the power buses/ground.
 Power bus management: The supply voltages in a PC are often too unstable for
sensitive measurements.
o High-frequency noise from other parts
o Line voltage fluctuations
o Power supply architecture
 High voltages for analog functions ( 15V)
 Low voltages for digital subsystems ( 5V)
 dc/dc boost converter (5V to  20V)
 Linear regulator ( 20V to both analog and digital voltages)
o Recommendations:
 Linear regulators remove low-frequency noise.
 Since boost converters usually operate in the 100 KHz range, a simple
LPF between the converter and the linear regulator can remove highfrequency noise from the power supply.
 Power supply decoupling:
o Bulk capacitor: Large value tantalum devices.
o Parallel capacitor: A tantalum capacitor has an equivalent series resistance
(ESR) of 0.5 to 1  at high frequencies – a ceramic capacitor, because of its
negligibly low ESR, connected in parallel often provides better decoupling.
o Local capacitor: It provides a low-impedance path to ground for power line ac
noise. A local capacitor should be as close as possible to the IC’s power-input
pin in order to minimize the coupling area between it and the IC pin.
o A typical bypass circuit:


Digital/analog interface:
o When digital-control signals are passed to analog components, switching
noise from the digital circuit supply can be spread to the analog circuits.
o Although digital circuits are often immune from the switching noise, analog
circuits can be adversely affected by it.
o Buffering the digital-control signal: A clean (heavily filtered) 5V rail is used
exclusively to power the buffers.
Grounding:
o Poorly designed grounding schemes: Ground loops and series injection
o Good grounding structures:
 Separate ground planes for the voltage source
 Single-point connections among the planes.
Analog
Digital
Boost
Converter

o If this is not possible:
 No more than one connection between the analog and the digital areas.
 No more than one connection between the boost converter and the
digital areas.
o More grounding tips:
 The best place to join the analog and the digital ground planes is at the
A/D converter since it is where the measured analog data are
converted to digital signals.
 Place the power supplies within the ground planes of the circuits they
power to confine the returning currents.
Interference: Potential difference between ground planes
o A small inductor between ground plane helps.
o The bulk capacitor discussed above forms an LC LPF with Q = (1/R) (L/C)1/2
 For flat response, Q < 0.707
 For a 100 F with ESR = 0.5 , L = 12.5 H.
 Wire size (current requirement)
 In high-frequency (high-speed) lines signal return through the least
inductive path (least loop area), which is usually the path directly
under the signal trace.
 Do not disturb the return flow or the return current may travel in an
unexpected path and causes ground loop noise.




EMI shielding:
o EMI in PCs (monitors, video cards, other plug-in peripherals) in the form of
magnetic fields is usually beyond our control.
o Shielding: Cost usually prohibits shielding the entire instrument.
o Shielding materials used must work at both high and low frequencies.
o Shielding effectiveness: reflective capability (conductivity) and absorption
capability (permeability).
o Steel and nickel both make good shields.
Shielding effectiveness:
o Absorption loss A  3.34 t f  R  R where t is the thickness of the shield, f is
the frequency,  R is the relative conductivity, and  R is the relative
permeability. For examples, A = 100, 100, and 1000 db for a 0.76 mm thick
steel shield at 100, 10K, and 1 MHz, respectively.
o To avoid current loops a good shield must provide a current path and should
be grounded at only one point.
o Because a ground is mainly needed for the protection of sensitive analog
systems, it should be grounded to the digital ground plane.
o Ideally, PCB should have shields on both sides.
o A single shield on the circuit side and ground planes on or within the PCB are
usually enough.
Proper circuit layout: Shielding is a quick and easy way to reduce EMI. However,
there is no substitute for proper circuit layout.
o Eliminate/minimize large current loop
o Place components as close together as possible
o Avoid long leads
o Use ground planes
Temperature:
o 5 to 15 C hotter inside
o Reduced air flow
o Temperature gradient (change of T)
o On-board reference for calibration