Pulse Shaping Filter and Matched Filter

Introduction

In a digital communication system, digital information can be sent on a carrier through changes in its fundamental characteristics such as: phase, frequency, and amplitude. In a physical channel, these transitions can be made smooth by implementing different types of transmit filters. In fact, the use of a filter plays an important part in a communications channel because it is effective at eliminating spectral leakage, reducing channel width, and eliminating interference from adjacent symbols (Inter Symbol Interference, ISI).In bandlimited channels, intersymbol interference (ISI) can be caused by multi-path fading as signals are transmitted over long distances and through various mediums. More specifically, this characteristic of the physical environment causes some symbols to be spread beyond their given time interval. As a result, they can interfere with the following or preceding transmitted symbols. One solution to this problem is the application of the pulse shaping filter . By applying this filter to each symbol that is generated, we are able to reduce channel bandwidth while reducing ISI. In addition, it is common to apply a match filter on the receiver side to minimize these affects.

Pulse Shaping Filter

In communications systems, two important requirements of a wireless communications channel demand the use of a pulse shaping filter. These requirements are: 1) generating bandlimited channels, and 2) reducing inter symbol interference (ISI) from multi-path signal reflections. Both requirements can be accomplished by a pulse shaping filter which is applied to each symbol. In fact, the sinc pulse, shown below, meets both of these requirements because it efficiently utilizes the frequency domain to utilize a smaller portion of the frequency domain, and because of the windowing affect that it has on each symbol period of a modulated signal. A sinc pulse is shown below along with an FFT spectrum of the given signal.

Time Domain and Frequency Domain spectrum of Pulse Shaping Filter 

Matched Filter

In signal processing, a matched filter is obtained by correlating a known signal, or template, with an unknown signal to detect the presence of the template in the unknown signal. This is 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 signal to noise ratio (SNR) in the presence of additive stochastic noise.It is well known, that the optimum receiver for an AWGN channel is the matched filter receiver.

The matched filter for a linearly modulated signal using pulse shape p(t)is shown below.

Matched filter schematic

The slicer determines which symbol is “closest” to the matched filter output.

Its operation depends on the symbols being used and the a priori probabilities.

while the pulse shaping filter serves the purpose of generating signals such that each symbol period does not overlap, the matched filter is important to filter out what signal reflections do occur in the transmission process. Because in multipath propogation model a direct-path signal arrives at the receiver before a reflected signal does, it is possible for the reflected signal to overlap with a subsequent symbol period. 

Complete matched filter

Raised Cosine/ Root Raised Cosine Filter

In signal processing, a root-raised-cosine filter (RRC), sometimes known as square-root-raised-cosine filter (SRRC), is frequently used as the transmit and receive filter in a digital communication system to perform matched filtering. This helps in minimizing intersymbol interference (ISI). To have minimum ISI (Intersymbol interference), the overall response of transmit filter, channel response and receive filter has to satisfy Nyquist ISI criterion. 

Mathematical equation used to define raised cosine filter is :

Mathematical expression for raised cosine filter

In this equation, α is the rolloff factor, which determines the sharpness of the frequency response. In addition, R is the number of samples per symbol.

Raised-cosine filter is the most popular filter response satisfying this criterion. Half of this filtering is done on the transmit side and half of this is done on the receive side. On the receive side, the channel response, if it can be accurately estimated, can also be taken into account so that the overall response is Raised-cosine filter.The combined response of two such filters is that of the raised-cosine (RC) filter.