Correlation is a foundation over which the whole structure of digital communications is built. In fact, correlation is the heart of a digital communication system, not only for data detection but for parameter estimation of various kinds as well. Throughout, we will find recurring reminders of this fact. As a start, consider from the article on Discrete Fourier Transform that each DFT output $S[k]$ is just a sum of term-by-term products between an input signal and a cosine/sine wave, which is actually a computation of correlation. Later, we will learn that to detect the transmitted bits at the receiver, correlation

Continue reading## Interpolation in Digital Communication Receivers

Timing synchronization in a digital receiver is about finding the right symbol peak and the symbol rate at which digital samples are taken for decisions purpose in a constellation diagram. In general, interpolation is the process of reproducing a missing sample at a desired location. In digital and wireless communications, the role of interpolation can be explained as follows. Background Imagine a Tx signal constructed from the upsampled and pulse shaped modulation symbols. The job of the Rx is to sample this waveform at optimal intervals, i.e., exactly at the middle of the eye diagram. In other words, the Rx

Continue reading## Singular Value Decomposition (SVD) – A Tutorial with an Application to Wireless Systems

Singular Value Decomposition (SVD) is a powerful concept in linear algebra whose relevance has significantly increased in recent times. Some of the notable examples are its applications in machine learning, data science and wireless communication systems. In this tutorial, I will explain the logic behind SVD from a non-mathematical viewpoint using a wireless application that forms the backbone of high speed wireless systems such as WiFi, 4G and 5G. What is Orthogonality and Why We Like It Orthogonality is a concept that comes with heavy mathematical details. However, it can be explained in a simple and non-rigorous manner. Look at

Continue reading## Pulse Amplitude Modulation (PAM)

In the article on modulation – from numbers to signals, we said that the Pulse Amplitude Modulation (PAM) is an amplitude scaling of the pulse $p(nT_S)$ according to the symbol value. What happens when this process of scaling the pulse amplitude by symbols is repeated for every symbol during each interval $T_M$? Clearly, a series of bits $b$ (1010 in our initial example) can be transmitted by choosing a rectangular pulse and scaling it with appropriate symbols. \begin{equation*} \begin{aligned} m = 0 \quad b = 1 \quad a[0] = +A \\ m = 1 \quad b = 0 \quad a[1]

Continue reading## Linear Systems

In wireless communications and other applications of digital signal processing, we often want to modify a generated or acquired signal. A device or algorithm that performs some prescribed operations on an input signal to generate an output signal is called a system. Amplifiers in communication receivers and filters in image processing applications are some systems that we interact with in daily lives. Our main focus in these articles will be on a particular class of systems which are linear and time-invariant. A linear system implies that if two inputs are scaled and summed together to form a new input, the

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