IIR Bandpass Filters Using Cascaded Biquads
This post provides a Matlab function that builds Butterworth bandpass IIR filters by cascading second-order biquad sections. The biquad approach, implemented in Direct Form II, reduces sensitivity to coefficient quantization, which matters most for narrowband filters. The included biquad_bp function computes each section's feedforward and feedback coefficients plus gains from a lowpass prototype order, center frequency, bandwidth, and sampling rate.
Controlling a DSP Network's Gain: A Note For DSP Beginners
Rick Lyons calls out a simple but costly mistake beginners make when normalizing digital networks, scaling the input instead of the output. Using fixed-point examples he shows that pre-multiplying an A/D output by 1/8 throws away bits and costs about 18 dB of SQNR. The practical guidance is to place gain control as the final multiplication stage and beware a faulty Simpson's 1/3 integrator example.
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.
Free Goodies from Embedded World - Full Inventory and Upcoming Draw Live-Streaming Date
Stephane came back from Embedded World with a massive haul of development kits, tools and swag and decided to give it away to multiple winners. Read the full inventory, learn how to enter by liking or sharing the LinkedIn and Twitter posts, and tune in Friday March 29 at 1pm EST on EmbeddedRelated.tv for the live draw where winners will pick their prizes.
Angle Addition Formulas from Euler's Formula
Complex numbers are rotations and scalings in the plane, and Cedron Dawg walks through polar and Cartesian representations to make that concrete. Using Euler's formula, the article shows how multiplying complex numbers multiplies magnitudes and adds angles, and how that directly yields the sine and cosine angle-addition formulas. Practical notes cover using atan2/arg and a brief Gambas example to verify results.
Demonstrating the Periodic Spectrum of a Sampled Signal Using the DFT
This post makes a basic DSP principle tangible by computing the DFT over an extended set of bins and plotting the results. It demonstrates that a sampled signal's spectrum repeats every sampling rate, explains the k-to-frequency mapping, and contrasts common bin ranges such as 0..N-1 and -N/2..N/2-1. The write-up also highlights symmetry for real sequences and recommends using the FFT for efficiency.
Free Goodies from Embedded World - What to Do Next?
Stephane Boucher went on a hunt for free stuff at Embedded World to assemble a giveaway bundle for a lucky reader. This short update shares that haul and asks the embedded community for ideas on what to do next. It is a conversational call for suggestions, aiming to turn conference swag into a useful prize.
Back from Embedded World 2019 - Funny Stories and Live-Streaming Woes
Stephane Boucher tried live-streaming multiple talks from Embedded World 2019 and turned a chaotic experiment into a useful set of lessons for embedded engineers. Between broken tripods, flaky venue WiFi, tricky German SIM purchases, and audio nightmares, he learned practical fixes for reliable streams and better video quality. Read this if you want candid, tactical advice on streaming hardware, connectivity, and on-site troubleshooting.
Spread the Word and Run a Chance to Win a Bundle of Goodies from Embedded World
EmbeddedRelated is turning Embedded World into a live-streaming experiment, and the more engineers help spread the word, the better the coverage could get. Stephane Boucher is asking readers to follow updates on Twitter and LinkedIn, where every like, share, or repost adds another chance to win a box of vendor goodies. The prize mix includes t-shirts, dev kits, gadgets, and plenty of pens.
Launch of EmbeddedRelated.tv
Stephane Boucher launches EmbeddedRelated.tv to host live broadcasts from Embedded World, starting next week. The site will show a constantly evolving schedule, a Live! tab to find ongoing streams, and ad-hoc demos added from the show floor. Expect schedule conflicts and small hiccups, and plan to refresh the page and join the forum thread for real-time updates and feedback.
An Efficient Linear Interpolation Scheme
A simple trick slashes the cost of linear interpolation to at most one multiply per output sample, and often to none. The post shows a zero-order-hold based network that preserves input samples, has a short L-1 transient, and lets 1/L scaling be implemented as a binary shift when L is a power of two. It also gives a fixed-point layout that moves scaling to the end to reduce quantization distortion.
Algebra's Laws of Powers and Roots: Handle With Care
Rick Lyons shows that familiar power and root rules from algebra can break down when exponents are complex. He tests common identities for two scenarios, real and fully complex exponents, with positive and negative mantissas, and compiles a table of cases that sometimes fail. The post includes MATLAB examples that reproduce counterexamples and a clear warning to numerically verify algebraic steps involving complex powers.
Resolving 'Can't initialize target CPU' on TI C6000 DSPs - Part 1
Misconfigured Code Composer Studio settings cause most 'Can't initialize target CPU' errors on TI C6000 boards, not a faulty silicon. Mike Dunn walks through the practical first steps: confirm your CCS version, identify the exact emulator and board or device part number, and ensure you have the correct emulator driver. The post also shows how to duplicate TI's factory board configuration to avoid common setup mistakes.
Improved Narrowband Lowpass IIR Filters
Rick Lyons presents a practical trick from his DSP book that makes narrowband lowpass IIR filters usable in fixed-point systems. By replacing unit delays with M-length delay lines to form an interpolated-IIR, pole radii and angles are transformed so desired poles fall into quantizer-friendly locations without wider coefficient words or extra multiplies. A following CIC image-reject stage removes replicated passbands to meet tight stopband specs.
Wavelets II - Vanishing Moments and Spectral Factorization
This post walks through how vanishing moments turn into concrete algebraic constraints on wavelet filter coefficients, and why that leads to Daubechies filters. It explains how a wavelet with A vanishing moments is orthogonal to all polynomials up to degree A minus one, and it shows how those continuous conditions become discrete sums like sum_k k^n h1(k)=0. Expect clear links between approximation power and filter length.
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.
Noise shaping
Markus Nentwig presents a compact, practical introduction to noise shaping by treating quantization error as the first sample of a designed impulse response. He shows how to derive a noise shaper from a target spectrum, demonstrates the tradeoff between in-band noise reduction and total noise increase, and includes a Matlab example while highlighting clipping and stability caveats for sigma-delta contexts.
A Recipe for a Basic Trigonometry Table
Cedron Dawg walks through building a degree-based sine and cosine table from first principles, showing both recursive and multiplicative complex-tone generators. The article highlights simple drift-correction tricks such as mitigated squaring and compact normalization, gives series methods to compute one-degree and half-degree values, and includes practical C code that ties the table to DFT usage and frequency estimation.
Dealing With Fixed Point Fractions
Fixed-point fractional math is easy to botch, and this post lays out pragmatic ways to avoid those mistakes. It clarifies the difference between integer and fractional overflow, shows how Q notation helps track binary-point scaling, and explains why multiplies add sign bits that may require shifting. Read for concrete FPGA strategies: keeping bit growth, selective shifts, or aggressive normalization, plus testing tips.
A Fast Guaranteed-Stable Sliding DFT Algorithm
Rick Lyons presents a compact, computationally efficient sliding DFT that computes a single N-point DFT bin output for each input sample in real time. The design replaces the traditional complex resonator with a 2nd-order real resonator and uses pole/zero cancellation to match the DFT bin response. Crucially, the resonator poles remain on the z-plane unit circle even with quantized coefficients, guaranteeing numerical stability.
TCP/IP interface (Matlab/Octave)
Markus Nentwig supplies a compact set of mex C functions that let you control Ethernet-enabled measurement instruments directly from Matlab or Octave on Windows. The code opens raw TCP/IP sockets, sends SCPI commands, and handles ASCII and binary replies including binary-length headers. It intentionally avoids instrument-control toolboxes and timeouts for simplicity, and includes instrIf_socket, instrIf_write, instrIf_read and instrIf_close with simple usage examples.
Digital PLL's, Part 3 -- Phase Lock an NCO to an External Clock
Phase-locking a numerically controlled oscillator to an external clock that is unrelated to system clocks is practical and largely unexplored. Neil Robertson presents a time-domain digital PLL that converts the ADC-sampled clock into I/Q with a Hilbert transformer and measures phase error with a compact complex phase detector. The post shows loop-filter coefficient formulas and simulations that reveal how ADC quantization and Gaussian clock noise map into NCO phase noise and how loop bandwidth shapes the result.
Ten Little Algorithms, Part 6: Green’s Theorem and Swept-Area Detection
Jason shows how Green's Theorem becomes a practical, low-cost method to detect real-time rotation from two orthogonal sensors by accumulating swept area. The post derives a compact discrete integrator S[n] = S[n-1] + (x[n]*(y[n]-y[n-1]) - y[n]*(x[n]-x[n-1]))/2, compares integer and floating implementations, and analyzes noise scaling and sampling rate tradeoffs. Includes Python demos and threshold guidance.
Understanding Radio Frequency Distortion
Markus Nentwig breaks down how analog RF nonlinearities appear in a complex baseband model so you can simulate and predistort real transmitters. The article shows that even-order terms vanish in-band under narrowband assumptions, while odd-order products collapse to |BB(t)|^(n-1) BB(t) and do not depend on the carrier frequency. It also explains bandwidth scaling and includes a MATLAB example plus measured PA coefficients.
Angle Addition Formulas from Euler's Formula
Complex numbers are rotations and scalings in the plane, and Cedron Dawg walks through polar and Cartesian representations to make that concrete. Using Euler's formula, the article shows how multiplying complex numbers multiplies magnitudes and adds angles, and how that directly yields the sine and cosine angle-addition formulas. Practical notes cover using atan2/arg and a brief Gambas example to verify results.
Waveforms that are their own Fourier Transform
Steve Smith admits a long-standing mistake and overturns the claim that only Gaussians are their own Fourier transform. He gives trivial and nontrivial examples, explains why infinitely many such waveforms exist, and shows a quick discrete construction using the DFT with a 1/sqrt(N) normalization. Engineers get an intuitive 30-second argument plus a practical recipe to build self-Fourier signals.
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.
DAC Zero-Order Hold Models
This article provides two simple time-domain models of a DAC’s zero-order hold. These models will allow us to find time and frequency domain approximations of DAC outputs, and simulate analog filtering of those outputs. Developing the models is also a good way to learn about the DAC ZOH function.
Using the DFT as a Filter: Correcting a Misconception
Some sources claim the DFT, when used as a filter, shifts spectral energy down to DC. Rick Lyons shows that this is not true for consecutive DFT-bin outputs and explains the cause of the confusion: the FIR interpretation requires reversing the usual twiddle-factor order. He derives the DFT-bin frequency response, shows the bandpass center at 2πm/N, and explains when decimation does produce a translation to zero Hz.
5G NR QC-LDPC Encoding Algorithm
Lyons Zhang breaks down the 5G NR QC-LDPC encoding structure, emphasizing the circulant permutation matrices and why QC-LDPC suits simple hardware. The post defines the Q(P) and Q(-1) notation, summarizes BG1 and BG2 dimensions and selection rules, and shows how the parity-check matrix is partitioned for practical encoder implementations.
