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Pulse Shaping Filter

In order for digital signals to be transmitted over a channel, pulse shaping must be performed in the baseband section of the transmitter to convert 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 is "0", and if the signal level is negative, it is "1".
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However, the frequency spectrum of the rectangular pulse signal is infinitely wide, so distortion occurs when transmitted 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 over a channel with limited bandwidth.
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  • When one symbol reaches its maximum amplitude, the amplitudes of other symbols are exactly 0, meaning that symbols do not interfere with each other, achieving no 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:
    |500
    The sampling decision at the receiving end is as follows:
    |500

2 Baseband Filter#

2.1 Ideal Low-Pass Filter#

To shape the pulse into a sinc waveform, simply input a 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:
|500
As long as the sending interval of the sinc pulse signal is set to $\frac{1}{B}$, which corresponds to a symbol transmission rate of $R_B=2B$, inter-symbol interference can be achieved.

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:

H(f)={12B,0f<(1α)B14B{1+cosπ2Bα[fB(1α)]},(1α)Bf<(1+α)B0,f(1+α)B\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:
image
The unit impulse response of the raised cosine roll-off filter is:

h(t)=F1[H(f)]=sinc(2Bt)cos(2παBt)1(4αBt)2\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:
image
At this time, the filter's bandwidth 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:
image
At this time, the filter's bandwidth is $(1+\alpha)B=2B$.
When using the raised cosine roll-off filter for pulse shaping, to achieve no inter-symbol interference, the time interval between pulse signals must be $\frac{1}{2B}$, that is, the symbol rate is $R_B=2B$.
Since the raised cosine roll-off filter will widen 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.

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