Articles | Page 16 of 37

January 20, 2022 Wireless and SDR

Least Mean Square (LMS) Equalizer – A Tutorial

The LMS algorithm was first proposed by Bernard Widrow (a professor at Stanford University) and his PhD student Ted Hoff (the architect of the first microprocessor) in the 1960s. Due to its simplicity and robustness, it has been the most widely used adaptive filtering algorithm in real applications. An LMS equalizer in communication system design is just one of those beautiful examples and its other applications include noise and echo cancellation, beamforming, neural networks and so on. Background The wireless channel is a source of severe distortion in the received (Rx) signal and our main task is to remove the

Signal frequency and analysis frequency are not the same, resulting in DFT leakage and spectrum sampling away from the peak. The spectrum shown here is oversampled by 3 through 3x zero-padded FFT

October 16, 2026 DSP

Peak Interpolation for Sub-Bin Accuracy in Radar and Communications

In signal processing applications, one frequently encounters the problem of estimating the frequency of a complex sinusoid embedded in noise. For example, a DSP engineer will encounter this problem in the delayed echo of an FMCW radar that directly translates into target range, or in carrier frequency estimation of a wireless communication system after timing synchronization has already been achieved (see Chapter 6 for details of how modulation removal exposes the underlying complex sinusoid). Beyond communications and radar, this frequency estimation is also heavily relied upon in digital pitch tracking, speech synthesis, active and passive sonar detection, Doppler ultrasound and

Odd symmetry around frequency points at half symbol rate adding up to a flat spectrum

February 29, 2024 DSP / Wireless and SDR

Proof of Poisson Sum Formula

The Poisson sum formula was discovered by the French mathematician and physicist Siméon Denis Poisson. It has several applications in digital signal processing, among which our concern is the periodic summation of modulated pulses in digital communication systems. Assume that $p(t)$ is a pulse shape (or any continuous-time function if you are not familiar with digital communications) and $P(f)$ is its Fourier Transform. The pulse is sampled at a rate of $f_s$ to produce its discrete version $p(nT_s)$ where $T_s=1/f_s$ is the duration between two samples. The Poisson summation formula relates these two quantities as \begin{equation}\label{equation-poisson-sum-formula} \frac{1}{T_s}\sum _{k=-\infty}^{\infty} P\left(f+\frac{k}{T_s}\right) =

Intuition behind multiplying two complex numbers

March 14, 2025 Wireless and SDR

Intuitive Reason behind Multiplication and Division of Complex Numbers

Any introduction to complex numbers and their operations follows a common pattern: the formulas are given without building any intuition. If you have ever wondered about why $j=\sqrt{-1}$, you can read about the origin of complex numbers. In this article, I will explain the intuitive reason behind why a product of two complex numbers multiplies their magnitudes and adds their angles, and a division of two complex numbers divides their magnitudes and subtracts their angles. Let us start with the multiplication. The division scenario can be analogously derived with inverse operations. Multiplication of Complex Numbers The intuition behind multiplying two

December 31, 2023 Wireless and SDR

Design of a Low-SNR Receiver

Wireless communication is energy inefficient due to the nature of the medium that spreads out energy in an unguided manner, as opposed to guided media like optical fiber and coaxial cable. To avoid wastage of power, one solution is to lower the transmit (Tx) power but then the receiver is left with the herculean task of efficiently demodulating the receive symbols at a low SNR. This article describes the design and implementation of one such receiver. Background The physical layer of a receiver system consists of three major parts, namely the frontend, the inner receiver, and the outer receiver. The