Key focus: Inter-symbol interference: symbols sent through a dispersive channel, arrive at different time intervals and interfere due to non-constant group delay.
Introduction
Communication systems have progressed from analog to digital implementation due to the latter’s advantages of bandwidth efficiency and exceptional immunity to noise. The greatest challenge to a communication systems engineer lies in designing a system that strikes a trade-off between the physical limitations of a given communication system, the need for higher capacity and the available resources such as bandwidth usage and power. Modern communication systems should operate in a very limited radio spectrum with minimized interference to other systems.
Numerous standardization committees define spectral masks for specific applications with the aim of reducing the interference to other systems by limiting the out-of-band radiations. Spectral masks are implemented by pulse shaping filters. Pulse shaping should contain the transmit signal within the specified band while minimizing the probability of errors at the receiver. On the other hand, pulse shaping causes inter-symbol interference (ISI) that degrades the detection process and therefore brings down the error performance of the whole system.
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Most of the communication channels (data storage devices, optical, wireless channels etc…) can be considered as band-limited linear filters. Thus these channels can be modeled as having the following frequency response
where, 
is amplitude response and 
is the phase response of the channel. The envelope or group delay for the given filter model is defined as,
A channel is considered non-distorting (within the given bandwidth 
occupied by the transmitted signal), when the amplitude response is constant and the phase response is a linear function of frequency. In other words, the channel has a constant group delay.
Amplitude distortion occurs when the amplitude response is no longer a constant and delay distortion or phase distortion occurs when the phase response is not a linear function of frequency (that is, the envelope or group delay is not a constant). Channels with delay distortion are termed as dispersive channels.
When a succession of pulses are transferred through a dispersive channel, the pulses may arrive at different time intervals the output of the channel due to non-constant group delay. As a result, the transmitted pulses may interfere with each other, rendering them completely indistinguishable at the receiver. This phenomenon is called inter-symbol interference (ISI). Dispersive effects are particularly relevant in high data rate communication systems. Delay dispersion can also manifest in time-variant multipath channels, since the copies of signals traveling via each propagating path may arrive at different times at the receiver thereby giving rise to ISI. Following are the major approaches to deal with ISI.
- Nyquist first criterion: Force ISI effect to zero by signal design – pulse shaping techniques like sinc filtering, raised-cosine filtering and square-root raised-cosine filtering
- Partial response signaling: Introduce controlled amount of ISI in the transmit side and prepare to deal with it at the receiver
- Design algorithms to counter ISI: Learn to live with the presence of ISI and design robust algorithms at the receiver – Viterbi algorithm↗ (maximum likelihood sequence estimation), equalizer etc.,
Demo in Matlab and Python
Consider a band-limited channel, modeled as an ideal brickwall low pass filter. The cut-off frequency of the low pass channel is 
rad/sample. An unit impulse is sent through the channel. As expected, we get a sinc pulse at the channel output (Figure 1, plots in the first row).
Next, a series of pulses with values [1,-1,1] are sent through the band-limited channel, at low data rate. Low data rate implies, there is sufficient pauses between the transmission of each value. The output of the channel shows clearly distinguishable peaks. As a consequence, the receiver can distinguish the values by simply sampling at the signal peaks (sampling is done at regular interval). This scenario is illustrated by Figure 1 row 2 plots.
Finally, plots in the row 3 of Figure 1 is for high speed data transmission, where the values [1,-1,1] are transmitted consecutively without any pause in between them. Due to the band-limited nature of the dispersive channel, the sinc pulses interfere with each other, thereby producing a smeared output. Now, the receiver cannot discern the values from the received signal. This phenomenon is called ‘inter-symbol interference’ (to be exact – inter-bit interference in this case 🙂 ).
Download the Python program for this demo – ipython notebook (as pdf).↗
Download the Matlab program for this demo.↗
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Topics in this chapter
| Pulse Shaping, Matched Filtering and Partial Response Signaling ● Introduction ● Nyquist Criterion for zero ISI ● Discrete-time model for a system with pulse shaping and matched filtering □ Rectangular pulse shaping □ Sinc pulse shaping □ Raised-cosine pulse shaping □ Square-root raised-cosine pulse shaping ● Eye Diagram ● Implementing a Matched Filter system with SRRC filtering □ Plotting the eye diagram □ Performance simulation ● Partial Response Signaling Models □ Impulse response and frequency response of PR signaling schemes ● Precoding □ Implementing a modulo-M precoder □ Simulation and results |
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