# Pulse Shaping in Digital communication

I am new here. I hope I can get help.

I want to ask a question regarding digital communication.

Does anyone know what are pulse shaping used for? Are they used for ISI mitigation or for something else? Also what is the receiver side of this operation is it matched filter?

As you've told the main goal of pulse shaping filter is to restrict signal bandwidth in the transmitter while in receiver you should place the matched filter to maximize SNR. In the practical modems pulse shaping filter in the transmitter and matched filter in the receiver are usually the same in terms of frequency response. The task is to choose filters to minimize ISI (not all TX-RX filters pairs satisfy this condition while if even maximizing SNR value). From the theory is known raised cosine filter satisfy ISI condition. You can read about this filter here. Let's place such a filter in TX and RX side:

$H_{system} = H_{Tx} * H_{Rx}$

We can see the total response of the system is no longer raised cosine. So the root raised cosine filter was proposed. The main idea is

$H_{rrc} = \sqrt{H_{rc}}$, so for our system

$H_{system} = H_{Tx} * H_{Rx} = H_{rrc} * H_{rrc} = (\sqrt{H_{rc}})^2 = H_{rc}$

By doing this we can minimize ISI and maximize SNR if the filter bandwidth is closely matched to signal's one. So it is very practical and widely used in digital modems (especially single carriers (SC) modems).

The main reason for using pulse shaping is that the bandwidth of the transmitted signal is the same as that of your pulse shape.

To avoid ISI, you want your pulse shape to be orthogonal to time-shifted versions of itself. What this means is that, if your pulse shape is $p(t)$ and you're transmitting at rate $T_s$ pulses per second, you want $$\int_{-\infty}^{\infty} p(t-kT_s)p(t-lT_s)\,dt$$ to be 0 for all integers $k$ and $l$, $k\neq l$. It is convenient to choose $p(t)$ such that the integral is 1 when $k=l$.

If you transmit $Ap(t)$, then the received signal is $$r(t)=Ap(t)+n(t),$$ where $n(t)$ is random Gaussian noise. If you assume $p(t)$ is symmetrical (as is almost always the case in practice), then the matched filter has impulse response $p(t)$, and its output is $A+n,$ where $n$ is a Gaussian random variable of variance $\sigma_n^2$. The importance of the matched filter is that, among all possible filters, it maximizes the signal-to-noise ratio (SNR) $A^2/\sigma_n^2$. Since the bit error rate (BER) depends on the SNR, the matched filter minimizes the BER too.

• So basically we use pulse shaping to shape our input signal that is going to be sent of the channel, such that ISI is mitigated? – Henry Apr 24 '15 at 20:56
• Yes. The reasons boil down to: * shaping your spectrum and bandwidth; * eliminating ISI; *allow use of a matched filter in the receiver. – MBaz Apr 24 '15 at 20:59