Filtering is a commonly used strategy for enhancing the signal to noise ratio. The fundamental aspects are intimately connected with the analysis of signals and time invariant signals. Consider the transfer function of the simple RC circuit used as an example of a time invariant system. This can be written as
If one considers the autocorrelation function of a noise signal with the substitution u = t + t,
then R(t) is essentially equivalent to the convolution of the functions x(t) and y(t), where y(t) = x(-t). Using the fact that for this special relationship the Fourier transform of y(t) satisfies Y(w) = X*(w), the relationship
follows from the convolution theorem and the fact that the transform of the autocorrelation function is the power spectrum. Consider the situation of a noise signal with power spectrum Sin(w) at the input to a linear time invariant system with transfer function H(w). Then using the general relation Yout(w) = H(w)Yin(w) the relation between the power spectrum of the noise at the output to that at the input is
Thus the quantity controlling the noise power is the square of the magnitude of the transfer function. For the particular case of the RC circuit under discussion,
Clearly this transfer function is unity at DC or zero frequency and tends to zero at frequencies which are large compared to the inverse of the time constant of the circuit. This is easily understood physically from the properties of the impedance of the capacitor, which acts like an open circuit for zero frequency and as a dead short at infinite frequency. As can be seen from the diagram below the effect of the transfer function is to leave unaltered those components with frequencies significantly less than the characteristic frequency 1/t, while strongly attenuating those with much higher frequencies. Because of this behaviour the circuit is referred to as a low pass filter, indicating that low frequency components are passed while high frequency components are filtered.
This formula holds both for the input power spectrum Sin(f) and the output power spectrum, |H(f)|2Sin(f). For simple rectangular filters encompassing the entire signal, the signal-to-noise ratio satisfies
Although different circuits have different transfer functions and hence exhibit other behaviours, in general the frequency dependence of the transfer function implies a frequency-dependant attenuation. For this reason the systems are generally referred to as filters. They are classified into three main categories, low pass, high pass and band pass filters. The second type displays complementary behaviour to the low pass filter and attenuates low frequency components while passing unaltered very high frequencies.
The band pass filter is more selective, passing with minor attenuation a range of frequencies known as the pass band. Frequency components either below or above the pass band are strongly attenuated. A rather different and more specialized filter is the notch filter which behaves in a complementary fashion to the band pass filter, strongly attenuating components in a band while passing those above and below the band.
Since noise is widely distributed in frequency, filtering always reduces the noise contribution. As can be seen from Eqn(11.6),for white noise the rms amplitude varies with the square root of the bandwidth. If the signal, or more importantly, the information in the signal, is concentrated in a range of frequencies then a filter with pass band matched to the signal frequency range will produce an output with enhanced signal to noise ratio.
The variety of transfer functions which may be produced with actual filters is limited by the frequency behaviour of the circuit elements. Digital methods on the other hand have no such limitation and represent a powerful method for signal enhancement, a procedure referred to as digital signal processing. Activity in this field is of such a level to warrant publication of a journal devoted entirely to, and bearing the title of, digital signal processing (DSP). In this approach a transient analyser is used to capture the raw signal, ie the output of front end hardware such as an amplifier containing the desired signal and the noise. The signal may then be processed computationally, using any appropriate algorithm. The process may be convolution in the time domain. A simple example is smoothing. Here a signal point is replaced by the average of a group of points in the region. In the smallest possible case of a three point smooth, the value is replaced by the average obtained from the point and its immediately adjacent neighbours. This procedure is the digital realization of convolution with a rectangular response function. In more complex cases it is often advantageous to do the processing in the frequency domain employing fast Fourier transform (FFT) algorithms. In any case when the processing is the mathematical analogue of the behaviour of a linear time invariant filter, the system is referred to as a digital filter. Because this approach is not limited by hardware constructions, more sophisticated approaches bringing to bear the power of statistical estimation theory may be employed. This field is one of active research and undergoing continual development.