Partial response (PR) signaling Model

Consider the generic baseband communication system model and its equivalent representation, shown in Figure 1, where the various blocks in the system are represented as filters. To have no ISI at the symbol sampling instants, the equivalent filter $latex H(f)$ should satisfy Nyquist’s first criterion.

If the system is ideal and noiseless, it can be characterized by samples of the desired impulse response $latex h(t)$. Let’s represent all the $latex N$ non-zero sample values of the desired impulse response, taken at symbol sampling spacing $latex T_{sym}$, as $latex {q_n}$, for $latex n=0,1,2,\cdots,N-1$.

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The partial response signaling model, illustrated in Figure 2, is expressed as a cascaded combination of a tapped delay line filter with tap coefficients set to $latex {q_n}$ and a filter with frequency response $latex G(f)$. The filter $latex Q(f)$ forces the desired sample values. On the other hand, the filter $latex G(f)$ bandlimits the system response and at the same time it preserves the sample values from the filter $latex Q(f)$ . The choice of filter coefficients for the filter $latex Q(f)$ and the different choices for $latex G(f)$ for satisfying Nyquist first criterion, result in different impulse response $latex H(f)$, but renders identical sample values $latex b_n$ in Figure 2 [1].

To have a system with minimum possible bandwidth, the filter $latex G(f)$ is chosen as

The inverse Fourier transform of $latex G(f)$ results in a sinc pulse. The corresponding overall impulse response of the system $latex h(t)$ is given by

If the bandwidth can be relaxed, other ISI free pulse-shapers like raised cosine can be considered for the $latex G(f)$ filter.

Given the nature of real world channels, it is not always desirable to satisfy Nyquist’s first criterion. For example, the channel in magnetic recording, exhibits spectral null at certain frequencies and therefore it defines the channel’s upper frequency limit. In such cases, it is very difficult to satisfy Nyquist first criterion. An alternative viable solution is to allow a controlled amount of ISI between the adjacent samples at the output of the equivalent filter $latex Q(f)$ shown in Figure 2. This deliberate injection of controlled amount of ISI is called partial response (PR) signaling or correlative coding.

Partial Response Signaling Schemes

Several classes of PR signaling schemes and their corresponding transfer functions represented as $latex Q(D)$ (where $latex D$ is the delay operator) are shown in Table 1. The unit delay is equal to a delay of 1 symbol duration ($latex T_{sym}$) in a continuous time system.

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References

[1] Peter Kabal and Subbarayan Pasupathy, Partial-response signaling, IEEE Transactions on Communications, Vol. 23, No. 9, pp. 921-934, September 1975.↗

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