0.10 Pulse shaping and receive filtering  (Page 9/13)

The raised cosine pulse $h_{RC}\left(t\right)$ with nonzero $\beta$ has the following characteristics:

• zero crossings at desired times,
• band edges of $H_{RC}\left(f\right)$ that are less severe than with a sinc pulse,
• an envelope that falls off at approximately $1/{\left|t\right|}^3$ for large $t$ (look at [link] ). This is significantly faster than $1/\left|t\right|$ . As the rolloff factor $\beta$ increases from 0 to 1, the significant part of the impulse response gets shorter.

Thus, we have seen several examples of Nyquist pulses: rectangular, Hamming, sinc, and raised cosine with avariety of roll off factors. What is the general principle that distinguishesNyquist pulses from all others? A necessary and sufficient conditionfor a signal $v\left(t\right)$ with Fourier transform $V\left(f\right)$ to be a Nyquist pulse is that the sum (over all $n$ ) of $V(f-n{f}_0)$ be constant. To see this, use the sifting property ofan impulse [link] to factor $V\left(f\right)$ from the sum:

Given that convolution in the frequency domain is multiplication in the time domain [link] , applying the definition of the Fourier transform,and using the transform pair (from [link] with $w\left(t\right)=1$ and $W\left(f\right)=\delta \left(f\right)$ )

where $f_0=1/T$ , this becomes

If $v\left(t\right)$ is a Nyquist pulse, the only nonzero term in the sum is $v\left(0\right)$ , and

Thus, the sum of the $V(f-n{f}_0)$ is a constant if $v\left(t\right)$ is a Nyquist pulse. Conversely, if the sum of the $V(f-n{f}_0)$ is a constant, then only the DC term in [link] can be nonzero, and so $v\left(t\right)$ is a Nyquist pulse.

Consider the 1.2 msec wide pulse shape $p\left(t\right)$ shown in [link] .

1. Is $p\left(t\right)$ a Nyquist pulse for the symbol period $T=0.35$ msec? Justify your answer.
2. Is $p\left(t\right)$ a Nyquist pulse for the symbol period $T=0.70$ msec? Justify your answer.

Intersymbol interference occurs when data values at one sample instant interfere with the data values at anothersampling instant. Using Nyquist shapes such as the rectangle, sinc, and raised cosine pulses removes the interference,at least at the correct sampling instants, when the channel is ideal. The next sections parlay this discussion ofisolated pulse shapes into usable designs for the pulse shaping and receive filters.