## How to Design Nyquist and Square-Root Nyquist Pulse Shaping Filters

The radio spectrum is a very precious resource like real estate and must be utilized judiciously. Pulse shaping filters control the spectral leakage of the transmitted signal in a wireless channel due to the strict restrictions to comply with a spectral mask. This is even more important for the upcoming 5G wireless systems which are based on a variety of wireless transmission protocols (such as mobile networks, Internet of Things (IoT) and machine to machine communications) combined in one comprehensive standard. Even for wired channels, there is always a natural bandwidth of the medium (copper wire, coaxial cable, optical fiber)

Quadrature Amplitude Modulation (QAM) is a spectrally efficient modulation scheme used in most of the high-speed wireless networks today. We discussed earlier that Pulse Amplitude Modulation (PAM) transmits information through amplitude scaling of the pulse $p(nT_S)$ according to the symbol value. To understand QAM, two routes need to be traversed. Route 1 We start the first route with differentiating between baseband and passband signals. A baseband signal has a spectral magnitude that is nonzero only for frequencies around origin ($F=0$) and negligible elsewhere. An example spectral plot for a PAM waveform is shown below for 500 2-PAM symbols shaped by

In digital logic, a stream of 1s and 0s forms a sequence of rectangular pulses, which can be easily identified at the receiver side by a threshold. In time domain, everything looks nice and perfect. Let us investigate the system characteristics in frequency domain. In a Pulse Amplitude Modulation (PAM) system, the main component that defines the spectral contents of the signal is the pulse shape $p(nT_S)$ at the Tx. We start with our attention towards a simple rectangular pulse shape. Here is a brief outline of what we cover in this article. Table of Contents 1. Spectrum of a