A discrete-time integrator implemented through a forward difference and a backward difference technique
DSP

Discrete-Time Integrators

An integrator is a very important filter that proves useful in implementation of many blocks of a communication receiver such as symbol timing synchronization and Phase-Locked Loop (PLL). It is an inverse operation to a differentiator that is also used in many signal processing applications such as FM demodulation and image processing. In continuous-time case, an integrator finds the area under the curve of a signal amplitude. A discrete-time system deals with just the signal samples and hence a discrete-time integrator serves the purpose of collecting a running sum of past samples for an input signal. Looking at an infinitesimally

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Three different cases for carrier frequency offset
Wireless and SDR

What is Carrier Frequency Offset (CFO) and How It Distorts the Rx Symbols

In Physics, frequency in units of Hz is defined as the number of cycles per unit time. Angular frequency is the rate of change of phase of a sinusoidal waveform with units of radians/second. \begin{equation*} 2\pi f = \frac{\Delta \theta}{\Delta t} \end{equation*} where $\Delta\theta$ and $\Delta t$ are the changes in phase and time, respectively. A Carrier Frequency Offset (CFO) usually arises due to two reasons. The video below also explains this concept. [Frequency mismatch between the Tx and Rx oscillators] No two devices are the same and there is always some difference between the manufacturer’s nominal specification and the

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Frequency response of a moving average filter
DSP

Moving Average Filter

The most commonly used filter in DSP applications is a moving average filter. In today’s world with extremely fast clock speeds of the microprocessors, it seems strange that an application would require simple operations. But that is exactly the case with most applications in embedded systems that run on limited battery power and consequently host small microcontrollers. For noise reduction, it can be implemented with a few adders and delay elements. For lowpass filtering, an excellent frequency domain response and substantial suppression of stopband sidelobes are less important than having a basic filtering functionality, which is where a moving average

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OFDM slices the spectrum just like a bread
Wireless and SDR

A Beginner’s Guide to OFDM

In the recent past, high data rate wireless communications is often considered synonymous to an Orthogonal Frequency Division Multiplexing (OFDM) system. OFDM is a special case of multi-carrier communication as opposed to a conventional single-carrier system.  The concepts on which OFDM is based are so simple that almost everyone in the wireless community is a technical expert in this subject. However, I have always felt an absence of a really simple guide on how OFDM works which can prove useful for technical persons not wanting to deal with too much technicalities, such as DSP experts outside communications, computer programmers, ham

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Multipath components with large Doppler shifts compared to the signal bandwidth
Wireless and SDR

Slow and Fast Fading in Wireless Channels

We discussed the idea of fading in wireless channels in a previous article. To understand different types of fading in the context of time variations, refer to the figure below that shows a multipath channel. Slow Fading A slow motion scenario is illustrated in the figure below where three multipath components are arriving with Doppler shifts $F_{D,i}$ from the carrier frequency. In this scenario, the magnitudes of $F_{D,i}$ are small and hence observe very little spreading of the cumulative spectrum. This can be understood by recalling that when two sinusoids with two different frequencies $F_1$ and $F_2$ are added, the

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