## Designing a matched filter with ISI in mind |
In-depth

#### It can be shown (see in-depth section) that a detection filter will be 'matched' to the input symbol pulse shape if the filter is designed with an impulse response that is in fact a time reversed and delayed replica of the input symbol shape. Alternatively, the frequency domain response of a matched filter must be equal to the complex conjugate of the spectrum of the input symbol.

#### Clearly, implementing a matched filter requires detailed knowledge of the source data symbol shape, and also relies on the symbol shape remaining undistorted as it passes through the channel. It also only holds true for Additive White Gaussian Noise. |

#### If we can assume that the channel is distortion free or has been equalized to remove the distortion, then we can introduce the concept of a matched filter pair, with one filter performing the pulse shaping in the transmitter and the other performing matched detection in the receiver. We already have a requirement for paired filtering, however, driven by the desire to achieve zero intersymbol interference. The important question is: does the Nyquist filtering requirement for zero ISI conflict with the matched filtering requirement for optimum S/N ratio? Fortunately it has been proven mathematically (Schwartz, 1990) that the root raised cosine filter pair satisfies both criteria, which explains its popularity with modem designers around the world.