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:
Signal information
Noise information
Signal-to-noise (S/N) ratio
Distortion
Implementation
Signal information:
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
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:
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.

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.

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.

Other noise types – Man-make noise, atmospheric disturbances, and …

Signal-to-noise (S/N) ratio:
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.
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.
The gain should be zero (the loss should be infinite) outside the signal frequency band.
Distortion:
Maximum S/N ratio usually requires the system to have a non-uniform frequency response.
Non-uniform frequency responses introduce distortion because signal components at different frequencies are amplified (attenuated) differently.
A practical system has to consider the trade-offs between noise and distortion.

Implementation:
A physical system cannot provide infinite energy – an all-pass filter is impossible to build.
A physical system must be casual – discontinuities in the response spectrum cannot be realized.

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:
Background noise is assumed to be white noise or band-limited white noise with the noise bandwidth encompasses the entire signal spectrum.
The signal of interest is deterministic.
Background noise is assumed to be additive: Input = signal + noise

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.
Because of the non-uniform frequency response, the filter output is distorted.
This technique is ideal for signal detection since in this case only the signal energy (with respect to noise), not the actual shape, is.
This technique is not useful if the shape of the signal waveform is unknown.

Mathematical details:
From linear system theory the output is obtained from the convolution integral:

For a white noise input with unity magnitude the S/N ratio at t = T is:

Note: The inequality is the direct result of the Schwarz inequality. The equality holds if the following is true:

, which means: .

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:

For white noise process, the impulse response of the matched filter is:

Numerical examples:
Rectangular pulses:

Saw-tooth pulses:

Examples:
White noise process:

Realistic low-pass noise:

Radar echo detection:

Matched filter implementation

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:
The filter that minimizes the mean-square error (minimum distortion) has the following transfer function H( f ).

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.
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:

where has all of the LHP poles and zeros and has all of the RHP poles and zeros. Split the poles and zeros on the jaxis evenly to both planes.

2)Use partial fraction expansion:

where has all of the LHP poles and zeros and 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,

3)The transfer function of the casual Wiener filter is given by:

Example:

For a non-casual filter:

Step 1:

Step 2:

Step 3:

Appendix: Noise Suppression (Paul Packebush; How to keep instruments accurate inside hot, noisy PCs; EDN, 4-13, 2000, p. 189)

Introduction:
PCs are less expensive and more powerful now.
They are being used in more and more measurement and control systems.
Often, the instruments are actually inside the PC.
Within a PC’s hot, noisy environment accurate measurements of low voltage and high-frequency signals are difficult to make.
Both data-acquisition cards as well as digital components present problems.
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:
Temperature variation: 5 to 15C higher than the ambient temperature.
Power supply dissipation
Reduced airflow
Hot CPU & Peripherals
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).
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.
High-frequency noise from other parts
Line voltage fluctuations
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)
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 high-frequency noise from the power supply.
Power supply decoupling:
Bulk capacitor: Large value tantalum devices.
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.
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.
A typical bypass circuit:
Digital/analog interface:
When digital-control signals are passed to analog components, switching noise from the digital circuit supply can be spread to the analog circuits.
Although digital circuits are often immune from the switching noise, analog circuits can be adversely affected by it.
Buffering the digital-control signal: A clean (heavily filtered) 5V rail is used exclusively to power the buffers.
Grounding:
Poorly designed grounding schemes: Ground loops and series injection
Good grounding structures:
Separate ground planes for the voltage source
Single-point connections among the planes.

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.
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
A small inductor between ground plane helps.
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:
EMI in PCs (monitors, video cards, other plug-in peripherals) in the form of magnetic fields is usually beyond our control.
Shielding: Cost usually prohibits shielding the entire instrument.
Shielding materials used must work at both high and low frequencies.
Shielding effectiveness: reflective capability (conductivity) and absorption capability (permeability).
Steel and nickel both make good shields.
Shielding effectiveness:
Absorption loss where t is the thickness of the shield, f is the frequency, is the relative conductivity, and 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.
To avoid current loops a good shield must provide a current path and should be grounded at only one point.
Because a ground is mainly needed for the protection of sensitive analog systems, it should be grounded to the digital ground plane.
Ideally, PCB should have shields on both sides.
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.
Eliminate/minimize large current loop
Place components as close together as possible
Avoid long leads
Use ground planes
Temperature:
5 to 15 C hotter inside
Reduced air flow
Temperature gradient (change of T)
On-board reference for calibration