In this section we examine designs for the transmitter and receiver filters that shape the overall signal pulse-shape function so as to ideally eliminate interference between adjacent pulses.
This is formally stated as Nyquist’s criterion for zero ISI.
5.4.1 Pulses Having the Zero ISI Property
To see how one might implement this approach, we recall the sampling theorem, which gives a theoretical maximum spacing between samples to be taken from a signal with an ideal lowpass spectrum in order that the signal can be reconstructed exactly from the sample values.
In particular, the transmission of a lowpass signal with bandwidth𝑊 hertz can be viewed as sending a minimum of2𝑊 independent sps. If these2𝑊 sps represent2𝑊 independent pieces of data, this transmission can be viewed as sending2𝑊 pulses per second through a channel represented by an ideal lowpass filter of bandwidth𝑊. The transmission of the𝑛th piece of information through the channel at time𝑡 = 𝑛𝑇 = 𝑛∕ (2𝑊 ) is accomplished by sending an impulse of amplitude𝑎𝑛. The output of the channel due to this impulse at the input is
𝑦𝑛(𝑡) = 𝑎𝑛sinc [2𝑊 (
𝑡 − 𝑛2𝑊 )]
(5.33) For an input consisting of a train of impulses spaced by𝑇 = 1∕ (2𝑊 )s, the channel output is
𝑦(𝑡) =∑
𝑛
𝑦𝑛(𝑡) =∑
𝑛
𝑎𝑛sinc [2𝑊 (
𝑡 − 𝑛 2𝑊
)]
(5.34) where{
𝑎𝑛}
is the sequence of sample values (i.e., the information). If the channel output is sampled at time𝑡𝑚= 𝑚∕2𝑊, the sample value is𝑎𝑚because
sinc(𝑚 − 𝑛) =
{1, 𝑚 = 𝑛
0, 𝑚≠𝑛 (5.35)
which results in all terms in(5.34)except the𝑚th being zero. In other words, the𝑚th sample value at the output is not affected by preceding or succeeding sample values; it represents an independent piece of information.
Note that the bandlimited channel implies that the time response due to the𝑛th impulse at the input is infinite in extent; a waveform cannot be simultaneously bandlimited and time- limited. It is of interest to inquire if there are any bandlimited waveforms other than sinc(2𝑊 𝑡) that have the property of(5.35), that is, that their zero crossings are spaced by𝑇 = 1∕ (2𝑊 ) seconds. One such family of pulses are those having raised cosine spectra. Their time response is given by
𝑝RC(𝑡) = cos(𝜋𝛽𝑡∕𝑇 ) 1 − (2𝛽𝑡∕𝑇 )2sinc
(𝑡 𝑇
)
(5.36) and their spectra by
𝑃RC(𝑓) =
⎧⎪
⎪⎨
⎪⎪
⎩
𝑇 , |𝑓|≤ 1−𝛽2𝑇
𝑇2
{1 + cos[
𝜋𝑇𝛽
(|𝑓|− 1−𝛽
2𝑇
)]}, 1−𝛽2𝑇 <|𝑓|≤ 1+𝛽2𝑇
0, |𝑓|> 1+𝛽2𝑇
(5.37)
where 𝛽 is called the roll-off factor. Figure 5.7 shows this family of spectra and the corre- sponding pulse responses for several values of𝛽. Note that zero crossings for 𝑝RC(𝑡)occur at least every 𝑇 seconds. If 𝛽 = 1, the single-sided bandwidth of 𝑃RC(𝑓) is 𝑇1 hertz (just substitute𝛽 = 1into(5.37)), which is twice that for the case of𝛽 = 0[
sinc(𝑡∕𝑇 )pulse] . The price paid for the raised cosine roll-off with increasing frequency of𝑃RC(𝑓), which may be easier to realize as practical filters in the transmitter and receiver, is increased bandwidth.
1
1
0.5
0
–0.5 0.8 0.6 PRC (f)PRC (t)
Tf
t/T 0.4
0.2 0–1
–2 –1.5 –1 –0.5 0 0.5 1 1.5 2
–0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1
β = 0 β = 0.35 β = 0.7 β = 1
(a)
(b) Figure 5.7
(a) Raised cosine spectra and (b) corresponding pulse responses.
Also,𝑝RC(𝑡)for 𝛽 = 1 has a narrow main lobe with very low side lobes. This is advanta- geous in that interference with neighboring pulses is minimized if the sampling instants are slightly in error. Pulses with raised cosine spectra are used extensively in the design of digital communication systems.
5.4.2 Nyquist’s Pulse-Shaping Criterion
Nyquist’s pulse-shaping criterion states that a pulse-shape function 𝑝(𝑡), having a Fourier transform𝑃 (𝑓)that satisfies the criterion
∑∞ 𝑘=−∞
𝑃( 𝑓 + 𝑘𝑇
)= 𝑇 , |𝑓|≤ 1
2𝑇 (5.38)
results in a pulse-shape function with sample values 𝑝 (𝑛𝑇 ) =
{1, 𝑛 = 0
0, 𝑛≠0 (5.39)
Using this result, we can see that no adjacent pulse interference will result if the received data stream is represented as
𝑦(𝑡) =
∑∞ 𝑛=−∞
𝑎𝑛𝑝(𝑡 − 𝑛𝑇 ) (5.40)
and the sampling at the receiver occurs at integer multiples of𝑇 seconds at the pulse epochs.
For example, to obtain the𝑛 = 10th sample, one simply sets𝑡 = 10𝑇in(5.40), and the resulting sample is𝑎10, given that the result of Nyquist’s pulse-shaping criterion of(5.39)holds.
The proof of Nyquist’s pulse-shaping criterion follows easily by making use of the inverse Fourier representation for𝑝(𝑡), which is
𝑝(𝑡) =∫
∞
−∞𝑃 (𝑓) exp(𝑗2𝜋𝑓𝑡) 𝑑𝑓 (5.41)
For the𝑛th sample value, this expression can be written as
𝑝 (𝑛𝑇 ) =
∑∞ 𝑘=−∞∫
(2𝑘+1)∕2𝑇
−(2𝑘+1)∕2𝑇𝑃 (𝑓) exp(𝑗2𝜋𝑓𝑛𝑇 ) 𝑑𝑓 (5.42) where the inverse Fourier transform integral for 𝑝(𝑡) has been broken up into contiguous frequency intervals of length1∕𝑇Hz. By the change of variables𝑢 = 𝑓 − 𝑘∕𝑇,(5.42)becomes
𝑝 (𝑛𝑇 ) =
∑∞ 𝑘=−∞∫
1∕2𝑇
−1∕2𝑇𝑃( 𝑢 + 𝑘𝑇
)exp(𝑗2𝜋𝑛𝑇 𝑢)𝑑𝑢
=∫
1∕2𝑇
−1∕2𝑇
∑∞ 𝑘=−∞
𝑃( 𝑢 + 𝑘
𝑇
)exp(𝑗2𝜋𝑛𝑇 𝑢)𝑑𝑢 (5.43)
where the order of integration and summation has been reversed. By hypothesis
∑∞ 𝑘=−∞
𝑃 (𝑢 + 𝑘∕𝑇 ) = 𝑇 (5.44)
between the limits of integration, so that(5.43)becomes 𝑝 (𝑛𝑇 ) =∫
1∕2𝑇
−1∕2𝑇𝑇 exp(𝑗2𝜋𝑛𝑇 𝑢) 𝑑𝑢 = sinc (𝑛)
=
{1, 𝑛 = 0
0, 𝑛≠0 (5.45)
which completes the proof of Nyquist’s pulse-shaping criterion.
With the aid of this result, it is now apparent why the raised-cosine pulse family is free of intersymbol interference, even though the family is by no means unique. Note that what is excluded from the raised-cosine spectrum for|𝑓|< 𝑇1 hertz is filled by the spectral translate tail for|𝑓|> 𝑇1 hertz. Example 5.6 illustrates this for a simpler, although more impractical, spectrum than the raised-cosine spectrum.
EXAMPLE 5.6
Consider the triangular spectrum
𝑃Δ(𝑓) = 𝑇 Λ (𝑇𝑓) (5.46)
It is shown in Figure 5.8(a) and in Figure 5.8(b)∑∞ 𝑘=−∞𝑃Δ
(𝑓 + 𝑘𝑇)
is shown where it is evident that the sum is a constant. Using the transform pairΛ (𝑡∕𝐵)⟷𝐵sinc2(𝐵𝑓)and duality to get the transform pair𝑝Δ(𝑡) = sinc2(𝑡∕𝑇 )⟷𝑇 Λ (𝑇𝑓) = 𝑃Δ(𝑓), we see that this pulse-shape function does indeed have the zero-ISI property because𝑝Δ(𝑛𝑇 ) = sinc2(𝑛) = 0, 𝑛≠0,𝑛integer.
1
0.5
0 PΔ (Tf)/T
–5 –4 –3 –2 –1 0
Tf
1 2 3 4 5
1
0.5
ΣP (T(f–n/T)/TΔn 0
–5 –4 –3 –2 –1 0
Tf
1 2 3 4 5
(a)
(b) Figure 5.8
Illustration that a triangular spectrum (a), satisfies Nyquist’s zero-ISI criterion (b).
■
5.4.3 Transmitter and Receiver Filters for Zero ISI
Consider the simplified pulse transmission system of Figure 5.9. A source produces a sequence of sample values{
𝑎𝑛}
. Note that these are not necessarily quantized or binary digits, but they could be. For example, two bits per sample could be sent with four possible levels, representing 00, 01, 10, and 11. In the simplified transmitter model under consideration here, the 𝑘th sample value multiplies a unit impulse occuring at time𝑘𝑇 and this weighted impulse train is the input to a transmitter filter with impulse responseℎ𝑇(𝑡)and corresponding frequency response𝐻𝑇(𝑓). The noise for now is assumed to be zero (effects of noise will be considered in Chapter 9). Thus, the input signal to the transmission channel, represented by a filter having
Transmitter f ilter
Channel f ilter
x(t) y(t) v(t)
Receiver f ilter
Synchronization Sampler
Thres-
holder Data out Source
Noise +
Figure 5.9
Transmitter, channel, and receiver cascade illustrating the implementation of a zero-ISI communication system.
impulse responseℎ𝐶(𝑡)and corresponding frequency response𝐻𝐶(𝑓), for all time is 𝑥 (𝑡) =
∑∞ 𝑘=−∞
𝑎𝑘𝛿 (𝑡 − 𝑘𝑇 ) ∗ ℎ𝑇(𝑡)
=
∑∞ 𝑘=−∞
𝑎𝑘ℎ𝑇(𝑡 − 𝑘𝑇 ) (5.47)
The output of the channel is
𝑦 (𝑡) = 𝑥(𝑡) ∗ ℎ𝐶(𝑡) (5.48)
and the output of the receiver filter is
𝑣 (𝑡) = 𝑦(𝑡) ∗ ℎ𝑅(𝑡) (5.49)
We want the output of the receiver filter to have the zero-ISI property and, to be specific, we set
𝑣 (𝑡) =
∑∞ 𝑘=−∞
𝑎𝑘𝐴𝑝RC(
𝑡 − 𝑘𝑇 − 𝑡𝑑)
(5.50) where 𝑝RC(𝑡)is the raised-cosine pulse function,𝑡𝑑 represents the delay introduced by the cascade of filters, and𝐴represents an amplitude scale factor. Putting this all together, we have 𝐴𝑝RC(𝑡 − 𝑡𝑑) = ℎ𝑇(𝑡) ∗ ℎ𝐶(𝑡) ∗ ℎ𝑅(𝑡) (5.51) or, by Fourier-transforming both sides, we have
𝐴𝑃RC(𝑓) exp(−𝑗2𝜋𝑓𝑡𝑑) = 𝐻𝑇(𝑓)𝐻𝐶(𝑓)𝐻𝑅(𝑓) (5.52) In terms of amplitude responses this becomes
𝐴𝑃RC(𝑓) =||𝐻𝑇(𝑓)||||𝐻𝐶(𝑓)||||𝐻𝑅(𝑓)|| (5.53)
0 500 1000 1500 2000 2500 f, Hz
3000 3500 4000 4500 5000 1.8
Bit rate = 5000 bps; channel f ilter 3-dB frequency = 2000 Hz; no. of poles = 1 1.6
1.4 1.2 1
HR(f) or HT(f) 0.8 0.6 0.4 0.2 0
β = 0 β = 0.35 β = 0.7 β = 1
Figure 5.10
Transmitter and receiver filter amplitude responses that implement the zero-ISI condition assuming a first-order Butterworth channel filter and raised-cosine pulse shapes.
Now||𝐻𝐶(𝑓)||is fixed (the channel is whatever it is) and𝑃RC(𝑓)is specified. Suppose we want the transmitter and receiver filter amplitude responses to be the same. Then, solving (5.46) with||𝐻𝑇(𝑓)||=||𝐻𝑅(𝑓)||, we have
||𝐻𝑇(𝑓)||2=||𝐻𝑅(𝑓)||2= 𝐴𝑃RC(𝑓)
||𝐻𝐶(𝑓)|| (5.54)
or
||𝐻𝑇(𝑓)||=||𝐻𝑅(𝑓)||= 𝐴1∕2𝑃RC1∕2(𝑓)
||𝐻𝐶(𝑓)||1∕2 (5.55)
This amplitude response is shown in Figure 5.10 for raised-cosine spectra of various roll-off factors and for a channel filter assumed to have a first-order Butterworth amplitude response. We have not accounted for the effects of additive noise. If the noise spectrum is flat, the only change would be another multiplicative constant. The constants are arbitrary since they multiply both signal and noise alike.