MATCHED FILTERS

a general representation for a matched filter

The PSD of the additive input noise, n(t) is known.•determine the filter characteristics (impulse response or transfer function) that maximize the instantaneous power output with respect to the average output noise power, i.e. design to the criterion

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PSD of the input noise

•the design criterion minimizes the probability of error for received digital signals.

The linear filter that meets the design criterion is a matched filter.

The transfer function is:

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•A matched filter is obtained by correlating a known signal, with an unknown signal to detect the presence of the known signal in the unknown signal

•equivalent to convolving the unknown signal with a conjugated time-reversed version of the template.•The matched filter is the optimal linear filter for maximizing the (SNR) in the presence of additive stochastic noise.

The integrate and dump filter.

•convolution is an operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions, giving the area overlap between the two functions as a function of the amount that one of the original functions is translated

•Convolution of 2 square pulses, the resulting waveform is a triangular pulse.•One of the functions g(t) is first reflected at 0

And then offset by t making it )( tg

The area under the resulting product gives the convolution at t.The horizontal axis is t for f*g and for f and g

•For binary digital signalling, where the bit shape is rectangular, the realization of the matched filter becomes equivalent to an integrate and dump filter shown

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•We have designed an optimised filter ie one that permits us to sample the incoming signal at the instant when the signal level is maximum.

We must integrate the noise-corrupted input signal over one bit period, and then literally dump the output of the integrator at the end of the period.The timing of the start and end of the integration range or interval requires the use of an external bit synchronizer Expand the illustration by including this circuit component.Due to the presence of some remnant noise even after filtering, the output is not yet binary. We can make it binary by feeding it into a comparator at point C .

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Correlation Processing

A specially designed antenna transmits a short burst of radio wave energy in a selected direction. If the propagating wave strikes an object, such as the helicopter in this illustration, a small fraction of the energy is reflected back toward a radio receiver located near the transmitter. The transmitted pulse is a specific shape that we have selected, such as the triangle shown in this example

The received signal will consist of two parts: (1) a shifted and scaled version of the transmitted pulse, and (2) random noise, wherefrom?Since radio signals travel at a known rate, the speed of light, the shift between the transmitted and received pulse is a direct measure of the distance to the object being detected.

Correlation is a mathematical operation that is very similar to convolution.uses two signals to produce a third signal. This third signal is called the cross-correlation of the two input signals. If a signal is correlated with itself, the resulting signal is instead called the autocorrelation.

We correlate (another form of electronic integration) the processor input, r(t), with the known (reference) signal wave shape, s(t)

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Differential pulse code modulation (DPCM)

is a procedure of converting an analogue into a digital signal in which an analogue signal is sampled and then the difference between the actual sample value and its predicted value is quantized and then encoded forming a digital value.

(predicted value is based on previous sample or samples)

In one realisation or design we can use a set of tapped delay modules in which the tap outputs are multiplied by gain factors. We then sum the resulting products to produce the filtered output. This structure is called the transversal filter equaliser

Then design problem involves simply finding the best values for the tap gains for reducing ISI to zero on the output waveform. One application of the transversal filter is the matched filter