The DFT Output and Its Dimensions
The DFT gives N outputs for N samples, yet for real-valued signals most of those outputs are redundant. This post explains how conjugate symmetry organizes the output into a real DC bin, N/2-1 complex positive-frequency bins, a real Nyquist bin for even N, and then the conjugate mirror bins. A 64-point example illustrates which bins carry unique information and which can be discarded.
Amplitude modulation and the sampling theorem
I am working on the 11th and probably final chapter of Think DSP, which follows material my colleague Siddhartan Govindasamy developed for a class at Olin College. He introduces amplitude modulation as a clever way to sneak up on the Nyquist–Shannon sampling theorem.
Most of the code for the chapter is done: you can check it out in this IPython notebook. I haven't written the text yet, but I'll outline it here, and paste in the key...
Exponential Smoothing with a Wrinkle
Cedron Dawg shows how pairing forward and backward exponential smoothing produces exact, frequency-dependent dampening for sinusoids while canceling time-domain lag. The average of the two passes scales the tone by a closed-form factor, and their difference acts like a first-derivative with a quarter-cycle phase shift. The post derives the analytic dampening formulas, compares them to the derivative, and includes a Python demo for DFT preprocessing.
Differentiating and integrating discrete signals
Think DSP's new chapter digs into discrete differentiation and integration, using first differences, convolution, and FFTs to compare time and frequency domain views. The author reproduces diff via convolution then explores cumsum as its inverse and runs into two puzzling mismatches: noisy FFT amplitude ratios for nonperiodic data, and a time-domain convolution that does not reproduce cumsum for a sawtooth despite matching frequency responses. The post includes IPython notebooks and invites troubleshooting.
Discrete-Time PLLs, Part 1: Basics
In this series of tutorials on discrete-time PLLs we will be focusing on Phase-Locked Loops that can be implemented in discrete-time signal proessors such as FPGAs, DSPs and of course, MATLAB.
Compressive Sensing - Recovery of Sparse Signals (Part 1)
The amount of data that is generated has been increasing at a substantial rate since the beginning of the digital revolution. The constraints on the sampling and reconstruction of digital signals are derived from the well-known Nyquist-Shannon sampling theorem...
Summary of ROC Rules
This is a very short guide on how to find all possible outcomes of a system where Region of Convergence (ROC) and the original signal is not known.
Analytic Signal
In communication theory and modulation theory we always deal with two phases: In-phase (I) and Quadrature-phase (Q). The question that I will discuss in this blog is that why we use two phases and not more.
Multilayer Perceptrons and Event Classification with data from CODEC using Scilab and Weka
For my first blog, I thought I would introduce the reader to Scilab [1] and Weka [2]. In order to illustrate how they work, I will put together a script in Scilab that will sample using the microphone and CODEC on your PC and save the waveform as a CSV file.
Maximum Likelihood Estimation
Any observation has some degree of noise content that makes our observations uncertain. When we try to make conclusions based on noisy observations, we have to separate the dynamics of a signal from noise.
Generating Partially Correlated Random Variables
Designing signals to match a target covariance is simpler than it sounds. This post shows how to build partially correlated complex signals by hand for the two-signal case, then generalizes to N signals using the Cholesky decomposition. Short MATLAB examples demonstrate the two-line implementation and the article highlights numerical caveats when a covariance is only positive semidefinite.
Frequency Translation by Way of Lowpass FIR Filtering
Rick Lyons shows how you can translate a signal down in frequency and lowpass filter it in a single operation by embedding cosine mixing values into FIR coefficients. The post explains how to build the translating FIR, how to choose the number of coefficient sets, and how decimation can dramatically reduce storage needs while noting practical constraints like the requirement that ft be an integer submultiple of fs.
Multi-Decimation Stage Filtering for Sigma Delta ADCs: Design and Optimization
A Matlab toolbox streamlines the design and optimization of multi-stage decimation filters for sigma-delta ADCs. MSD-toolbox automates stage-count and decimation-factor selection, generates Parks-McClellan equiripple FIR coefficients, and iteratively selects coefficient quantization to meet in-band noise constraints. It accepts sigma-delta bitstream stimuli for spectral and intra-stage analysis, includes cost estimation routines, and is published open-source on MathWorks with examples and a dissertation reference.
Time Machine, Anyone?
Causal filters can look like time machines, but they do not break physics. Andor Bariska reproduces a classic electronic experiment in MATLAB, showing how a minimum-phase peaking filter and its FDLS biquad approximation produce negative group delay bands that make predictable, bandlimited signals appear to emerge early. The post walks through group delay, discretization, pulse and random-signal tests, and why unpredictability restores causality.
Feedback Controllers - Making Hardware with Firmware. Part 3. Sampled Data Aspects
This article digs into practical sampled-data issues you must address when building feedback controllers for circuit emulation. It highlights a common MATLAB versus Simulink discrepancy caused by DAC holding, explains why FOH (ramp-invariant) c2d conversion matters, and surveys latency, bit depth, filter and precision trade-offs. It also lists candidate ADCs, DACs and FPGAs used in a real evaluation platform to guide hardware choices.
Finding the Best Optimum
Optimization is seductive but often misleading, especially when mathematical models don't match messy reality. Tim Wescott shares stories from circuits and communications to show how chasing the theoretical global optimum can waste time and money. He recommends framing 'best' in practical terms, validating models, and optimizing for cost and impact so products ship on time and actually work in the real world.
Online DSP Classes: Why Such a High Dropout Rate?
Rick Lyons digs into a startling statistic: online DSP courses reported a 97% dropout rate. He argues the main culprits are math-heavy curricula that overwhelm beginners and rigid, non-self-paced schedules that demand sustained 8-10+ hours per week. Rick urges course creators to rethink pacing and mathematical depth to improve completion rates and student engagement.
Off Topic: Refraction in a Varying Medium
Cedron Dawg derives a compact vector differential equation for a point particle moving through a smoothly varying refractive medium using the Euler-Lagrange variational method. By introducing a log refractive index called "fluff density," the paper expresses acceleration purely in terms of the fluff gradient and velocity, then explores curvature, superposition, and point-source capture radii with simple closed-form results.
Coupled-Form 2nd-Order IIR Resonators: A Contradiction Resolved
Rick Lyons resolves a long-standing confusion about the coupled-form 2nd-order IIR resonator by deriving its correct z-domain transfer function and explaining why textbooks can appear to contradict pole plots. He shows that with infinite precision the coupled and standard denominators match, but finite-bit quantization of rcos(Θ) and rsin(Θ) changes the z^-2 coefficient and shifts pole positions. Read to learn the correct H(z) to predict quantized behavior and when the coupled form outperforms the standard design.
Helping New Bloggers to Break the Ice: A New Ipad Pro for the Author with the Best Article!
Breaking the ice can be tough. Over the years, many individuals have asked to be given access to the blogging interface only to never post an article.
Some Thoughts on Sampling
Sampling's 1/Ts amplitude factor is not a paradox but a consequence of axis scaling and impulse density, once you view the units correctly. This post walks through impulse trains in continuous and discrete time, uses DFT examples and Parseval's relation, and shows how downsampling and time scaling produce the familiar spectral replicas and their amplitudes. The geometry of the axes resolves the confusion.
Filtering Noise: The Basics (Part 1)
How do you pull signals out of random noise? This post builds intuition from first principles for discrete-time white Gaussian noise and shows how simple linear FIR filtering (averaging) reduces noise. You’ll get derivations for the output mean, variance and autocorrelation, learn why the uniform moving-average minimizes noise under a unity-DC constraint, and why its sinc spectrum can be problematic. Part 1 of a short series.
FIR sideways (interpolator polyphase decomposition)
Markus Nentwig presents a compact way to implement a symmetric FIR interpolator by rethinking the usual tapped delay line. The 1:3 polyphase example uses separate delay lines per coefficient to skip multiplies on known zeros and exploit symmetry, cutting multiplications substantially; a Matlab/Octave demo and notes on ASIC-friendly implementation are included to help evaluate real-world cost tradeoffs.
Helping New Bloggers to Break the Ice: A New Ipad Pro for the Author with the Best Article!
Breaking the ice can be tough. Over the years, many individuals have asked to be given access to the blogging interface only to never post an article.
Generating Partially Correlated Random Variables
Designing signals to match a target covariance is simpler than it sounds. This post shows how to build partially correlated complex signals by hand for the two-signal case, then generalizes to N signals using the Cholesky decomposition. Short MATLAB examples demonstrate the two-line implementation and the article highlights numerical caveats when a covariance is only positive semidefinite.
Multiplierless Half-band Filters and Hilbert Transformers
This article provides coefficients of multiplierless Finite Impulse Response 7-tap, 11-tap, and 15-tap half-band filters and Hilbert Transformers. Since Hilbert transformer coefficients are simply related to half-band coefficients, multiplierless Hilbert transformers are easily derived from multiplierless half-bands.
Add the Hilbert Transformer to Your DSP Toolkit, Part 2
This post shows a simple practical route to a Hilbert transformer by starting from a half-band FIR filter and tweaking its symmetry. It walks through a 19-tap example synthesized with Matlab's firpm (Parks-McClellan), explains the required frequency scaling, and shows how even-numbered taps become (or can be forced) zero through symmetry and coefficient quantization. Useful design rules are summarized for choosing ntaps.
An Efficient Full-Band Sliding DFT Spectrum Analyzer
Rick Lyons shows two compact sliding DFT networks that compute the 0th bin and all positive-frequency outputs for even and odd N, running sample-by-sample on real input streams. The designs reduce computational workload versus a prior observer-based sliding DFT by using fewer parallel paths, while remaining guaranteed stable and avoiding the traditional comb delay-line. A simple initialization and streaming procedure makes them practical for real-time spectrum analysis.
The Freshers Interview Guide
Hiring managers see the same avoidable mistakes from new grads, so Jeff offers blunt, practical advice to fix them. This short guide explains why honesty, solid debugging skills, and clear resumes matter more than cramming technical facts, and shows how to demonstrate problem-solving, organization, and teamwork in an interview to stand out as a reliable entry-level DSP or EE candidate.
Model Signal Impairments at Complex Baseband
Neil Robertson presents compact complex-baseband channel models for common signal impairments, implemented as short Matlab functions of up to seven lines. Using QAM examples and constellation plots, he demonstrates how interfering carriers, two-path multipath, sinusoidal phase noise, and Gaussian noise distort constellations and affect MER. The examples are lightweight and practical, making it easy to test receiver diagnostics and prototype adaptive-equalizer scenarios.
