# Documentation

## Filtering

The Communications System Toolbox™ software includes several functions, objects, and blocks that can help you design and use filters. Other filtering capabilities are in the Signal Processing Toolbox™ and the DSP System Toolbox™. The sections of this chapter are as follows:

For an example involving raised cosine filters, type ```showdemo rcosdemo```.

### Filter Features

Without propagation delays, both Hilbert filters and raised cosine filters are noncausal. This means that the current output depends on the system's future input. In order to design only realizable filters, the `hilbiir` function delays the input signal before producing an output. This delay, known as the filter's group delay, is the time between the filter's initial response and its peak response. The group delay is defined as

`$-\frac{d}{d\omega }\theta \left(\omega \right)$`

where θ represents the phase of the filter and ω represents the frequency in radians per second. This delay is set so that the impulse response before time zero is negligible and can safely be ignored by the function.

For example, the Hilbert filter whose impulse is shown below uses a group delay of one second. In the figure, the impulse response near time 0 is small and the large impulse response values occur near time 1.

Filtering tasks that blocks in the Communications System Toolbox support include:

• Filter Using Simulink Raised Cosine Filter Blocks. Raised cosine filters are very commonly used for pulse shaping and matched filtering. The following block diagram illustrates a typical use of raised cosine filters.

• Shaping a signal using ideal rectangular pulses.

• Implementing an integrate-and-dump operation or a windowed integrator. An integrate-and-dump operation is often used in a receiver model when the system's transmitter uses an ideal rectangular-pulse model. Integrate-and-dump can also be used in fiber optics and in spread-spectrum communication systems such as CDMA (code division multiple access) applications.

Additional filtering capabilities exist in the Filter Designs and Multirate Filters libraries of the DSP System Toolbox product.

For more background information about filters and pulse shaping, see the works listed in the Selected Bibliography Filtering.

### Selected Bibliography Filtering

[1] Korn, Israel, Digital Communications, New York, Van Nostrand Reinhold, 1985.

[2] Oppenheim, Alan V., and Ronald W. Schafer, Discrete-Time Signal Processing, Englewood Cliffs, NJ, Prentice Hall, 1989.

[3] Proakis, John G., Digital Communications, 3rd ed., New York, McGraw-Hill, 1995.

[4] Rappaport, Theodore S., Wireless Communications: Principles and Practice, Upper Saddle River, NJ, Prentice Hall, 1996.

[5] Sklar, Bernard, Digital Communications: Fundamentals and Applications, Englewood Cliffs, NJ, Prentice Hall, 1988.