Pulse Shaping Filter - moerjielovecookie

To transmit digital signals over a channel, pulse shaping must be performed in the baseband section of the transmitter, converting the digital signal into a pulse signal; after the pulse signal reaches the receiver, the digital signal is recovered through sampling and decision-making in the baseband section.

1 Pulse Shaping#

1.1 Rectangular Pulse#

The easiest pulse waveform to implement is the rectangular pulse. Taking the digital signal "00010110" as an example, at the transmitting end, "0" can be mapped to a positive pulse, and "1" can be mapped to a negative pulse. When sampling at the receiving end, if the signal level is positive, it corresponds to "0", and if the signal level is negative, it corresponds to "1".
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However, the frequency spectrum of the rectangular pulse signal is infinitely wide, so distortion occurs during transmission over a channel with limited bandwidth, which may even lead to sampling decision distortion, making it impossible to recover the digital signal.

1.2 Sinc Pulse#

The Sinc pulse signal has two advantages:

The frequency spectrum bandwidth of the Sinc signal is limited, so there is no distortion when transmitted through a channel with limited bandwidth.
When one symbol reaches its maximum amplitude, the amplitudes of other symbols are exactly zero, preventing inter-symbol interference.
Taking the digital signal 00010110 as an example, 0 is mapped to a positive pulse, and 1 is mapped to a negative pulse.
The waveform after pulse shaping at the transmitting end is as follows:

The sampling decision at the receiving end is as follows:

2 Baseband Filter#

2.1 Ideal Low-Pass Filter#

To shape the pulse into a sinc waveform, simply input the unit impulse signal into an ideal LPF to obtain the sinc pulse signal.
If the bandwidth of the LPF is B, the output sinc pulse signal waveform is as follows:
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As long as the transmission interval of the sinc pulse signal is set to $\frac{1}{B}$, which means the symbol transmission rate $R_B=2B$, inter-symbol interference can be avoided.

2.2 Raised Cosine Roll-off Filter#

The sinc pulse signal obtained by filtering the unit impulse signal with an ideal low-pass filter has a relatively large tail oscillation amplitude and a slow decay rate. When there is a timing deviation, inter-symbol interference can be significant. Considering that actual systems always have some timing errors, pulse shaping generally does not use an ideal low-pass filter but instead uses a raised cosine roll-off filter, which has a small tail amplitude and fast decay, beneficial for reducing inter-symbol interference and lowering timing requirements.
The frequency response of the raised cosine roll-off filter is:

\begin{aligned} &\mathrm{H}(f)=\begin{cases}\frac{1}{2B},&0\leqslant\left|f\right|<\left(1-\alpha\right)B\\\frac{1}{4B}\left\{1+\cos\frac{\pi}{2B\alpha}\Big[\left|f\right|-B\left(1-\alpha\right)\right]\Big\},&(1-\alpha)B\leqslant|f|<(1+\alpha)B\\0,&\left|f\right|\geqslant\left(1+\alpha\right)B \\ \end{cases}\end{aligned}

where $B=\frac{R_B}{2}$
The frequency response curve of the raised cosine roll-off filter is as follows:

The unit impulse response of the raised cosine roll-off filter is:

\mathrm h(t)=\mathscr{F}^{-1}\big[\mathrm H(f)\big]=\mathrm sinc\big(2Bt\big)\frac{\cos\big(2\pi\alpha Bt\big)}{1-\big(4\alpha Bt\big)^2}

where $\alpha$ is a very important parameter of the raised cosine roll-off filter, known as the roll-off factor.
When $\alpha=0$, the raised cosine roll-off filter is an ideal low-pass filter with bandwidth B.
When $\alpha=0.5$, the frequency response and unit impulse response of the raised cosine roll-off filter are as follows:

At this time, the bandwidth of the filter is $（1+\alpha）B=1.5B$.
When $\alpha=1$, the frequency response and unit impulse response of the raised cosine roll-off filter are as follows:

At this time, the bandwidth of the filter is $（1+\alpha）B=2B$.
When using the raised cosine roll-off filter for pulse shaping, to achieve inter-symbol interference avoidance, the time interval between pulse signals must be $\frac{1}{2B}$, meaning the symbol rate is $R_B=2B$.
Since the raised cosine roll-off filter will broaden the bandwidth, under a given symbol rate $R_B$, the frequency spectrum bandwidth of the baseband signal is $(1+\alpha)\times \frac{R_B}{2}$.

3 Eye Diagram#

The eye diagram can be used to evaluate the inter-symbol interference situation of a system.