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Transmit pulse shaping filter - rectangular and sinc (Nyquist)

Posted by Krishna Pillai
April 14, 2008

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

, where

is the symbol period,

is the symbol to transmit,

is the transmit filter,

is the symbol index and

is the output waveform.

In this post, the objective is to understand the properties of the transmit filter 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 .

The sequence of transmit symbols maybe visualized as follows.

transmission of sequence of baseband symbols

Figure: Transmit symbols for baseband PAM transmission

To recover the symbols from , one may sample the waveform at multiples of symbol interval . The sampled waveform can be,

.

Breaking the above equation into two parts,

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 should be zero at and non-zero at time .

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,

.

The filtered waveform can be as shown below.

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 .

Figure: Time domain response of sinc filter

As desired, the above filter has zeros at and non-zero value at time . The corresponding frequency response is a rectangular pulse bandlimited from Hz to Hz. The frequency response is as shown below:

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]).

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 symbols (with symbol period )isHz.

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.

Click here to download.

Matlab/Octave code for simulating transmit pulse shaping with rectangular and sinc waveforms

The observed spectrum is as shown below.

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 Hz to Hz. This is because, by theory, the sinc pulse exists from to . However, for simulations the sinc pulse is truncated for a finite duration. This results in negligible (less than -30dB) spectral content present outside 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].

Reference

[DIG-COMM-BARRY-LEE-MESSERSCHMITT] Digital Communication: Third Edition, by John R. Barry, Edward A. Lee, David G. Messerschmitt

Hope this helps.

Krishna

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    • Comments
    Comment by Yahia on April 21, 2008 @ 11:15 pm

    I have two questions!
    1- How do we decide the upsampling ratio before pulse shaping filter?
    2- How do we decide the length of pulse shape filter?

    Comment by Krishna on April 22, 2008 @ 7:40 am

    @Yahia:
    Both the q’s are related. One of the typical design constraint will be transmit spectral mask specification. I would expect the designer to choose a reasonable filter length with a reasonable upsampling ratio to meet the transmit mask specification with adequate margin. Further thoughts:

    1. Ideally one would want to have a high upsampling ratio, but is limited by practical considerations. Higher upsampling ratio means - DAC’s are clocked faster, the filter is clocked faster etc leading into hardware complexity. Typical upsampling ratio I ve seen is around 4x the symbol rate. Typically this should enable us to achieve adequate rejection of aliases with a reasonable filter length.

    2. As expected, more the filter length, better the rejection of aliases. However more taps means higher hardware complexity. As mentioned, above we may chose a filter sufficient to meet the mask with margin.

    Helps?

    Comment by Yahia on April 22, 2008 @ 8:58 am

    Thank you for your reply… I agree with you that most implementations use upsample ratio of 4. I guess that the reason for that to enable two levels of tracking [phase and frequency], if you upsample by 2 you will not be able to track the frequency drifts! Do you agree with me on this point?

    Your answer provided a great help. In fact I did not know that the main reference for deciding the filter order is the mask specification. That makes sense now to me. Thank you

    Comment by Krishna on April 23, 2008 @ 4:48 am

    @Yahia: hmm… I am not sure about your point on frequency cannot be tracked with upsampling factor of 2. I think the digital PLL should be able to track the frequency drifts. Let me think more and get back to you.

    Comment by Kelly Miller on April 30, 2008 @ 1:29 am

    Oh! Wonderful job!
    Very good and actual post.
    Thx, your blog in my RSS reader now

    Comment by umar on July 6, 2008 @ 9:39 pm

    Hi, i have a question about upsampling in your code.
    amUpSampled = [am;zeros(fs-1,length(am))]
    amU = amUpSampled(:).’
    are these lines basically upsampling the symbols by factor of 10.
    Regards,
    umar

    Comment by Krishna Pillai on July 7, 2008 @ 5:20 am

    @umar: Well, the upsampling by N involves two steps
    (a) first insert N-1 zeros between the samples, then
    (b) pass the zero inserted samples through a filter.
    The above lines capture step (a) i.e. inserting 9 zeros inbetween the samples.

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