## Design of a Discrete-Time Differentiator

Many signal processing algorithms require computation of the derivative of a signal in real-time. Some of the examples are timing recovery, carrier frequency synchronization, FM demodulation and demodulation of LoRa signals. An analog or digital filter that computes such a derivative is known as a differentiator. Before we design such a discrete-time differentiating filter, let us review some of the fundamentals. A Derivative The following quote is attributed to Heraclitus, a Greek philosopher, from 535 BC. Change is the only constant in life. This was brought into the realm of science by Newton and Leibniz. The purpose of science is

## Frequency Modulation (FM) and Demodulation Using DSP Techniques

Frequency Modulation (FM) is as old as the history of wireless communications itself. The past few decades saw the rise of digital signal processing in all spheres of life that pervaded even the implementation of analog modulation schemes. Today many of the FM systems are built using discrete-time techniques instead of the conventional circuitry as described below. Frequency Modulation In digital communications, data is sent through altering a characteristic of an electromagnetic wave such as amplitude, frequency or phase in discrete steps (e.g., $M$ number of levels). Such systems are known as Amplitude Shift Keying (ASK), Frequency Shift Keying (FSK)

## The Heterodyne Principle and the Superheterodyne Receiver

During World War I, Edwin Howard Armstrong invented the superheterodyne Rx as an alternative to the Tuned Radio Frequency (TRF) receivers that moved a tunable filter to the desired signal. His purpose was to overcome their limitations in regard to selectivity and sensitivity. To understand the principle of a heterodyne receiver, a pictorial representation is of utmost importance. While this is generally true for all concepts, there are specific issues of spectral translations in receiver architectures that require nice and clear figures. This is how I proceed below. The Heterodyne Principle Instead of employing a tunable bandpass filter that is