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Transmit pulse shaping filter – rectangular and sinc (Nyquist)

Posted By Krishna Sankar On April 14, 2008 @ 6:06 am In Filter | 20 Comments

In the previous post on I-Q modulator and de-modulator [1], we had briefly mentioned that the a baseband PAM transmission can be modelled as

$s(t) = \sum_{m=-\infty}^{\infty}a_mg(t-mT)$, where

$T$ is the symbol period,

$a_m$ is the symbol to transmit,

$g(t)$ is the transmit filter,

$m$ is the symbol index and

$s(t)$is the output waveform.

In this post, the objective is to understand the properties of the transmit filter$g(t)$ i.e. to find out a filter which occupies the minimum required bandwidth while ensuring inter-symbol-interference (ISI) free transmission of the information symbol $a_m$.

The sequence of transmit symbols maybe visualized as follows.

[2]

Figure: Transmit symbols for baseband PAM transmission

To recover the symbols from $s(t)$, one may sample the waveform at multiples of symbol interval $T$. The sampled waveform can be,

$s(kT)=\sum_{m=-\infty}^{+\infty}a_mg(kT-mT)$.

Breaking the above equation into two parts,

$s(kT)=g(0)a_k+\sum_{m \neq k}a_mg(kT-mT)$

From the above equation, it is intuitive that for ensuring no inter symbol interference (ISI), the second term in the above equation should be zero i.e. the taps of the filter $g(t)$ should be zero at $-T,\ T,\ 2T,\ldots$ and non-zero at time $0$.

Rectangular filter

The most simplest filter is the zero-order hold filter, i.e. to repeat the current symbol till the next symbol arrives and so on. Mathematically, the filter can be represented as,

$\begin{eqnarray}g(t)&=&1,\mbox{ } 0\le t < T\\&=& 0\mbox{ } \mbox{elsewhere}\end{eqnarray}$.

The filtered waveform can be as shown below.

[3]

Figure: Baseband PAM transmisison with rectangular filtering

Though there is no ISI with rectangular filtering, we will show later that this filtering is not be optimal from the bandwidth perspective.

Filtering with sinc() shaped pulses

From our textbooks, we may recall that sinc shaped pulses have have a band limited rectangular spectrum. For example, consider the sinc pulse of width $2T$.

$g(t) = \frac{sin(\pi t/T)}{\pi t/T},\mbox{ } t=-\infty \mbox{ to } +\infty$

Figure: Time domain response of sinc filter

As desired, the above filter has zeros at $-T,\ T,\ 2T,\ldots$ and non-zero value at time $0$. The corresponding frequency response is a rectangular pulse bandlimited from $-\frac{1}{2T}$Hz to $+\frac{1}{2T}$Hz. The frequency response is as shown below:

[4]

Figure: Frequency response of sinc shaped filter

Infact, the above sinc shaped pulse satisfies the Nyquist criterion and is called Nyquist pulse (Refer Sec 5.1.1 [DIG-COMM-BARRY-LEE-MESSERSCHMITT]). [5]

It follows that with the sinc shaped pulse (also called Nyquist pulse) used for transmit filtering:

(a) inter-symbol interference (ISI) is not introduced when sampled properly.

(b) minimum required bandwidth for transmitting $a_m$ symbols (with symbol period $T$)is$+\frac{1}{2T}$Hz.

Simulation model

Brief Matlab/Octave script for observing the spectrum of a random BPSK modulated symbols with rectangular filtering and sinc shape filtering might be helpful to uderstand the concept further.

The observed spectrum is as shown below.

[7]

Figure: Transmit spectrum with rectangular and sinc shaped pulse shaping filter

Summary

1. From the above spectrum plot, the sinc shaped filter does not result in perfectly bandlimited spectrum from from $-\frac{1}{2T}$Hz to $+\frac{1}{2T}$Hz. This is because, by theory, the sinc pulse exists from $-\infty$ to $+\infty$. However, for simulations the sinc pulse is truncated for a finite duration. This results in negligible (less than -30dB) spectral content present outside $\pm\frac{1}{2T}$Hz.

2. The above discussion does not present why the sinc shaped filtering is optimal for ensuring minimum bandwidth. For the theoretical understanding, one may refer Sec 5.1.1 of [DIG-COMM-BARRY-LEE-MESSERSCHMITT] [5].

Reference

Hope this helps.

Krishna

URL to article: http://www.dsplog.com/2008/04/14/transmit-pulse-shape-nyquist-sinc-rectangular/

URLs in this post:

[1] I-Q modulator and de-modulator: http://www.dsplog.com/2008/04/05/need-for-i-q-modulator-and-demodulator/