How do you design and implement Nyquist pulse shaping in digital communication?

1 Nyquist criterion

The Nyquist criterion states that the minimum sampling rate required to avoid aliasing is twice the highest frequency of the signal. This means that you can transmit up to 2B bits per second over a channel with bandwidth B, without losing any information. However, this assumes that the pulses are rectangular and have infinite duration, which is not practical. Therefore, you need to shape the pulses to make them finite and smooth, while still satisfying the Nyquist criterion.

2 Pulse shaping filters

Pulse shaping filters are devices that modify the shape of the pulses according to a desired impulse response. The impulse response determines how the filter affects the frequency and time domain characteristics of the signal. For example, a low-pass filter reduces the high-frequency components and makes the pulses smoother, but also increases their duration and causes ISI. A pulse shaping filter should balance these trade-offs and minimize the ISI.

3 Nyquist filters

Nyquist filters are a special class of pulse shaping filters that meet the Nyquist criterion. They have an impulse response that has zero crossings at the sampling instants, except for the peak at the center. This means that the pulses do not interfere with each other and can be recovered without distortion. Nyquist filters can have different shapes, such as raised cosine, root raised cosine, and Gaussian.

4 Designing Nyquist filters

To design a Nyquist filter, you need to specify some parameters, such as the roll-off factor, the filter order, and the sampling frequency. The roll-off factor determines how steeply the filter transitions from the passband to the stopband. A higher roll-off factor reduces the bandwidth of the signal, but also increases the ISI. The filter order determines how many coefficients the filter has and how complex it is. A higher filter order improves the performance of the filter, but also increases the computation and delay. The sampling frequency determines how many samples per second the filter operates on. A higher sampling frequency increases the resolution of the filter, but also requires more resources.

5 Implementing Nyquist filters

To implement a Nyquist filter, you need to use a digital signal processor (DSP) or a field-programmable gate array (FPGA) that can perform convolution operations. Convolution is a mathematical operation that applies the impulse response of the filter to the input signal and produces the output signal. You can use a software tool, such as MATLAB or Python, to design and simulate the filter and generate the coefficients. Then, you can program the DSP or FPGA to load the coefficients and perform convolution on the input signal. Alternatively, you can use a hardware module, such as a digital-to-analog converter (DAC) or a modulator, that has a built-in Nyquist filter and can shape the pulses directly.

7 Here’s what else to consider

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