Matched Filter - Example of Matched Filter in Digital Communications

Example of Matched Filter in Digital Communications

The matched filter is also used in communications. In the context of a communication system that sends binary messages from the transmitter to the receiver across a noisy channel, a matched filter can be used to detect the transmitted pulses in the noisy received signal.

Imagine we want to send the sequence "0101100100" coded in non polar Non-return-to-zero (NRZ) through a certain channel.

Mathematically, a sequence in NRZ code can be described as a sequence of unit pulses or shifted rect functions, each pulse being weighted by +1 if the bit is "1" and by 0 if the bit is "0". Formally, the scaling factor for the bit is,

We can represent our message, as the sum of shifted unit pulses:

where is the time length of one bit.

Thus, the signal to be sent by the transmitter is

If we model our noisy channel as an AWGN channel, white Gaussian noise is added to the signal. At the receiver end, for a Signal-to-noise ratio of 3dB, this may look like:

A first glance will not reveal the original transmitted sequence. There is a high power of noise relative to the power of the desired signal (i.e., there is a low signal-to-noise ratio). If the receiver were to sample this signal at the correct moments, the resulting binary message would possibly belie the original transmitted one.

To increase our signal-to-noise ratio, we pass the received signal through a matched filter. In this case, the filter should be matched to an NRZ pulse (equivalent to a "1" coded in NRZ code). Precisely, the impulse response of the ideal matched filter, assuming white (uncorrelated) noise should be a time-reversed complex-conjugated scaled version of the signal that we are seeking. We choose

In this case, due to symmetry, the time-reversed complex conjugate of is in fact, allowing us to call the impulse response of our matched filter convolution system.

After convolving with the correct matched filter, the resulting signal, is,

where denotes convolution.

Which can now be safely sampled by the receiver at the correct sampling instants, and compared to an appropriate threshold, resulting in a correct interpretation of the binary message.