# Use Nyquist Pulse Shaping for Zero Intersymbol Interference in Digital Communication Systems

### Key Takeaways

• Intersymbol interference (ISI) is a type of distortion resulting from dispersive communication channels. ISI can cause problems during the reception of signals.

• Pulse shaping is the process of choosing the time-domain and spectral shape of symbols so that pulses do not spread or overlap.

• Nyquist pulse shaping criterion introduced at the transmission end leads to improved signal reception without ISI.

The characteristics of communication channels play an important role in the reconstruction of original information at the receiver end

In digital communication, modulation converts discrete-time data pulses or symbols to continuous-time signals for transmission. A modulated signal is transmitted over the communication channel and demodulated at the receiving end to obtain the original information. The characteristics of a communication channel play an important role in the reconstruction of that original information. Ideally, communication channels would only scale and time-shift signals, however, in practical systems, channels introduce distortions into signals.

Intersymbol interference (ISI) is a type of distortion resulting from dispersive communication channels. ISI limits the error-free reception of signals. To mitigate this, engineers can use Nyquist pulse shaping techniques.

Nyquist pulse shaping criterion introduced in the transmission end leads to a more reliable reception of signals without ISI distortions. In this article, we will take a closer look at using Nyquist pulse shaping criterion to reduce intersymbol interference.

## The Frequency Response of Communication Channels

In digital communication, most communication channels are band-limited, and band-limited channels have finite bandwidth. They can be considered equivalent to band-limited linear filters.   Most of these channels are either baseband or passband.

The frequency response H(f) of communication channels can be given by equation 1 (below). The derivative of the phase response gives the group delay of the channel under consideration, as shown in equation 2 (below).

The frequency response of a baseband channel and passband channel is shown in the figure below. The baseband channel response is similar to a low pass filter, whereas the passband channel resembles that of a band-pass filter.

Baseband and passband channel frequency responses

From the amplitude and phase responses of a channel, we can conclude whether distortion is introduced into the signal transmitted. The communication channel can be considered distortionless, or a ‘non-distorting channel’, if A(f) is constant with frequency and 𝜙(f) is a linear function of frequency. However, in most cases, the amplitude response A(f) is not a constant, which causes amplitude distortions in the signal transmitted over the channel. If 𝜙(f) is not a linear function of frequency (non-constant group delay), it introduces phase distortion or delay distortions. The channels with delay distortions are called dispersive channels.

Non-distorting communication channel characteristics

## Intersymbol Interference and Why It Matters

In digital communication, pulses transmitted through dispersive channels reach the output side of the channel at different time intervals due to non-constant group delay. This results in intersymbol interference (ISI), which means interference between the adjacent transmitted pulses. The term ‘symbol’ is a synonym for data bits or pulses in digital communication. As the name suggests, the symbols overlap due to the dispersive characteristics of the communication channels.

ISI is prevalent in high data rate communication systems. When the communication channel bandwidth is greater than the signal bandwidth, ISI is minimized. When the bandwidth of the signal and the channel bandwidth is close to each other, then the overlapping of symbols is greater and ISI increases.

ISI is responsible for degrading the signal that is going to be transmitted. This makes the reconstruction of the original information at the receiving end difficult. In such cases, communication is hindered, which is why controlling ISI is so critical.

Transmitted sequence and received sequenced with ISI

## Nyquist Pulse Shaping

Pulse shaping is one technique used in communication to overcome the degradation of signals due to ISI. Pulse shaping involves the process of choosing the time-domain and spectral shape of the symbols so that the pulses do not spread or overlap.

Nyquist pulse shaping criterion is followed to achieve zero ISI. According to Nyquist pulse shaping criterion, to achieve zero ISI, digital pulses should satisfy the following conditions given in equation (3), where Tb is the bit period:

Pulses that satisfy the above conditions have zero ISI, as the values of the pulses at sampling periods are either equal to 1 at the center of the pulse or equal to zero at points where other pulses are centered. The input pulses shaped by following the Nyquist pulse shaping criterion result in the efficient reconstruction of original information at the receiver end. To achieve zero ISI, pulses can be shaped as sinc pulses, rectangular pulses, or raised cosine pulses. Various pulse shaping filtering schemes can be employed at the transmission end of communication systems to achieve a pulse shape that results in the perfect reconstruction of original signals, including sinc filtering, raised-cosine filtering, and square-root raised cosine filtering.