Two jobs
Stephane Boucher explains why EmbeddedRelated went quiet for a few months after a volunteer project demanded more of his time. He and his wife organized a clown-gymnastics show with 15 kids, sold more than 700 of 800 tickets, and raised $2,700 for the Tree of Hope. Now the shows are done and he plans to resume regular posting with new site features.
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.
Setting the 3-dB Cutoff Frequency of an Exponential Averager
Many engineers use a simple exponential averager but need the correct α to achieve a specified 3-dB cutoff. Rick Lyons compares a common approximation with the exact closed-form solution, shows when the approximation is valid, and derives the exact α in the appendix. The approximation works well for fc < 0.1fs, but it becomes noticeably inaccurate as the normalized cutoff increases.
'z' as in 'Zorro': Frequency Masking FIR
Markus Nentwig shows an efficient way to build steep wideband FIR filters by combining upsampled and complementary stages, then masking their spectra. He provides a Matlab and Octave design program that uses a generic least-squares optimizer to place coefficients, letting you explore filter sizes and oversampling while cutting computational cost significantly compared to a conventional symmetric FIR.
Do you like the new Comments System?
Stephane Boucher has just rolled out a new comments system for the DSPRelated blogs and wants feedback from readers. He’s asking the community to try it out, share thoughts, and help shake out any issues before it gets expanded to the code snippets and papers sections.
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.
Design of an anti-aliasing filter for a DAC
If you need a practical way to design an anti-aliasing filter for a DAC, this post delivers an Octave/Matlab script that numerically optimizes a Laplace-domain transfer function for linear phase and arbitrary magnitude. The routine models the DAC sample-and-hold sinc response, compensates group delay automatically, and can include an optional multiplierless FIR equalizer. An example shows a 5.4 dB objective improvement and reduced analog Q for easier implementation.
Understanding the 'Phasing Method' of Single Sideband Demodulation
Rick Lyons explains how the phasing method separates overlapping single sideband transmissions using quadrature processing and the Hilbert transform, making SSB demodulation practical in crowded RF environments. After reviewing simple synchronous detection, he walks through spectra and block diagrams that show how complex downconversion produces i and q paths which reinforce the desired sideband and cancel the other. The post also covers DSP implementation tips and BFO error effects.
Frequency-Domain Periodicity and the Discrete Fourier Transform
Sampling turns a continuous spectrum into an infinite set of replicas, and this article explains why the DFT and DTFT inevitably show periodic, circular spectra. Eric Jacobsen combines rigorous math with a geometric, wagon-wheel intuition to clarify aliasing, bandlimited sampling, and sampled-IF techniques. Read it to see when center frequency doesn't matter, how cyclic baseband shifts behave, and why bandwidth, not absolute frequency, determines alias-free sampling.
Time-Domain Periodicity and the Discrete Fourier Transform
Finite-length observation windows change how tones appear in a DFT, and Eric Jacobsen shows how the convolution theorem explains the familiar sin(x)/x main lobe and sidelobes. He contrasts two consistent viewpoints: viewing the DFT as a windowed signal convolved with the window transform, or as the transform of a periodically repeated sequence. Practical tips on zero-padding, bin spacing, and phase effects help avoid common misinterpretations.
A Two Bin Exact Frequency Formula for a Pure Complex Tone in a DFT
Cedron Dawg derives an exact two-bin frequency formula for a pure complex tone in the DFT, eliminating amplitude and phase to isolate frequency via a complex quotient and the complex logarithm. He presents an adjacent-bin simplification that replaces a complex multiply with a bin offset plus an atan2 angle, and discusses integer-frequency handling and aliasing. C source and numerical examples show the formula working in practice.
Stereophonic Amplitude-Panning: A Derivation of the 'Tangent Law'
Rick Lyons presents a clear geometrical derivation of the stereophonic amplitude-panning Tangent Law, filling a gap left by common references. Using vector components and the equidistant speaker assumption to keep signals in phase, he arrives at the Tangent Law and isolates practical gain formulas gL and gR needed to place an apparent source at a desired panning angle. Engineers can apply Eqs. (12) and (14) directly.
Fitting a Damped Sine Wave
Detlef Amberg presents a simple linear-algebra approach to recover frequency, phase, amplitude, and damping of a sampled damped sine wave. Instead of nonlinear fitting, the method casts the waveform as a second-order difference equation, uses linear regression to estimate b and omega, and recovers amplitude and phase by mixing with quadrature carriers; amplitude and damping are then fine-tuned with a gradient iteration. MATLAB code is available on File Exchange.
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.
'z' as in 'Zorro': Frequency Masking FIR
Markus Nentwig shows an efficient way to build steep wideband FIR filters by combining upsampled and complementary stages, then masking their spectra. He provides a Matlab and Octave design program that uses a generic least-squares optimizer to place coefficients, letting you explore filter sizes and oversampling while cutting computational cost significantly compared to a conventional symmetric FIR.
Feedback Controllers - Making Hardware with Firmware. Part 8. Control Loop Test-bed
Built around modest FPGA hardware, this post presents a practical test-bed for evaluating high-speed, low-latency feedback controllers. It covers ADC/DAC specifications, basic and arbitrary test signals, and an IFFT-based generator that can produce thousands of simultaneous tones for rapid Bode, phase, and latency measurements. The article also compares two IFFT strategies, explains turbo sampling, and shows open- and closed-loop test configurations.
An Efficient Lowpass Filter in Octave
Paul Lovell presents an efficient linear-phase lowpass FIR implemented in Octave, built as a Matrix IFIR with two matrix band-edge shaping stages followed by three recursive running-sum stages. The design reshapes input blocks into matrices to exploit interpolation structure and uses cumsum-based moving sums for speed. For a 200 Hz cutoff at 48 kHz the five-stage example ran about 15 times faster than a single-stage FIR.
Errata for the book: 'Understanding Digital Signal Processing'
Rick Lyons collects all errata for every edition and printing of his book Understanding Digital Signal Processing into one centralized list, with downloadable PDFs for each variant. The post also shows how to identify your book's printing number for American 1st, 2nd, and 3rd editions and flags a few oddball versions that lack errata.
Design Square-Root Nyquist Filters
A multirate signal processing textbook presents a neat method for designing square-root Nyquist FIR filters that combine zero ISI with strong stopband attenuation. This post walks through the principle that matched transmit and receive filters need square-root Nyquist responses, gives the key design relations for excess bandwidth and stopband edge, and includes a Matlab implementation to produce practical FIR matched filters for QAM-style systems.
FFT Interpolation Based on FFT Samples: A Detective Story With a Surprise Ending
Rick Lyons follows a numerical mismatch from a published astronomy paper into a short detective story about FFT interpolation. He shows a commonly published interpolation formula produces large errors, explains why the algebraic approximations fail, and presents several correct alternatives with algebraic simplifications that greatly reduce computation. Engineers get both the debugging lesson and practical, lower‑cost formulas for evaluating X(k) between FFT bins.
Compute Images/Aliases of CIC Interpolators/Decimators
CIC filters provide multiplier-free interpolation and decimation for large sample-rate changes, but their images and aliases can trip up designs. This post supplies two concise Matlab functions and hands-on examples to compute interpolator images and decimator aliases, showing spectra and freqz plots. Readers will learn how interpolation ratio and number of stages alter passband, stopband, and aliasing behavior.
Simple Discrete-Time Modeling of Lossy LC Filters
Converting a lossy LC filter into a discrete-time impulse response lets you analyze mixed analog and DSP systems in one time domain. This post walks through computing the LC frequency response via chain (ABCD) parameters including resistive losses, enforcing the Hermitian symmetry required for a real IDFT, and using the IDFT to produce an asymmetrical FIR impulse response. A 5th-order Butterworth example illustrates insertion loss and impulse-shape effects.
There's No End to It -- Matlab Code Plots Frequency Response above the Unit Circle
If you want a fresh way to inspect a digital filter, this post introduces plotfil3d, a compact MATLAB function that wraps the magnitude response around the unit circle in the Z-plane so you can view it in 3D. It uses freqz to compute H(z) in dB for N points and accepts an optional azimuth to change the viewing angle; the code is provided in the appendix.
Phase and Amplitude Calculation for a Pure Complex Tone in a DFT using Multiple Bins
Cedron presents exact, closed-form formulas to extract the phase and amplitude of a pure complex tone from multiple DFT bin values, using a compact vector formulation. The derivation introduces a delta variable to simplify the sinusoidal bin expression, stacks neighboring bins into a basis vector, and solves for the complex amplitude q by projection. The phase and magnitude follow directly from q, and extra bins reduce leakage when the tone falls between bins.
OpenCV for DSP/GPU, MSDN equivalent for CCS, and more
Porting OpenCV to DSPs could be a real business opportunity, but it is far from trivial, writes Shehrzad Qureshi. He highlights major obstacles: the engineering scale, mixed open-source licenses, and hard-to-parallelize primitives like connected components. He also criticizes Code Composer Studio's help system compared with MSDN, notes an ATI Stream talk, and announces a CUDA walkthrough on FFT-based image filtering.
The Discrete Fourier Transform of Symmetric Sequences
Symmetric sequences arise often in digital signal processing. Examples include symmetric pulses, window functions, and the coefficients of most finite-impulse response (FIR) filters, not to mention the cosine function. Examining symmetric sequences can give us some insights into the Discrete Fourier Transform (DFT). An even-symmetric sequence is centered at n = 0 and xeven(n) = xeven(-n). The DFT of xeven(n) is real. Most often, signals we encounter start at n = 0, so they are not strictly speaking even-symmetric. We’ll look at the relationship between the DFT’s of such sequences and those of true even-symmetric sequences.
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.
Accelerating Matlab DSP Code on the GPU
Seth Benton spent a few days testing Jacket to accelerate MATLAB on NVIDIA GPUs, and found it surprisingly easy to speed up DSP code. He ran 2D FFT and interp2 benchmarks on a MacBook Air with a GeForce 9400M, seeing impressive speedups for large images while hitting GPU memory and precision limits at high sizes. The post shares practical tips on casting to GPU types, minimizing CPU-GPU transfers, and when GPU acceleration is most useful.
How the Cooley-Tukey FFT Algorithm Works | Part 2 - Divide & Conquer
The Fast Fourier Transform revolutionized the Discrete Fourier Transform by making it much more efficient. In part 1, we saw that if you run the DFT on a power-of-2 number of samples, the calculations of different groups of samples repeat themselves at different frequencies. By leveraging the repeating patterns of sine and cosine values, the algorithm enables us to calculate the full DFT more efficiently. However, the calculations of certain groups of samples repeat more often than others. In this article, we’re going to explore how the divide-and-conquer method prepares the ground for the next stage of the algorithm by grouping the samples into specially ordered pairs.
DSPRelated and EmbeddedRelated now on Facebook & I will be at EE Live!
Stephane Boucher announces two practical updates for DSPRelated readers. He launched Facebook pages for DSPRelated and EmbeddedRelated so members can get faster updates, and he will be attending EE Live in San Jose from March 30 to April 3 with a $100-off promo code for early registration. He also asks the community for ideas on how to make his conference coverage most useful.
