Coefficients of Cascaded Discrete-Time Systems
Multiplying discrete-time transfer functions is just polynomial multiplication, and polynomial multiplication is convolution. Neil Robertson shows that the numerator and denominator coefficients of cascaded systems come from convolving the individual coefficient vectors, then demonstrates the idea with MATLAB code and a 2nd-order IIR cascade that yields a 4th-order response. The approach makes computing time and frequency responses straightforward.
Design IIR Filters Using Cascaded Biquads
High-order IIR filters are numerically sensitive, especially at low cutoff frequencies. This article shows how to implement a Butterworth lowpass as a cascade of second-order biquads, deriving the per-section coefficient formulas and giving a Matlab biquad_synth example. It explains computing denominator coefficients from pole pairs, using b = [1 2 1] with K = sum(a)/4 for unity DC gain, and highlights reduced quantization sensitivity.
Design IIR Highpass Filters
Neil Robertson walks through a compact, six-step procedure to synthesize IIR Butterworth highpass filters using pre-warping and the bilinear transform. The post gives the pole transformations, the placement of N zeros at z=1, the scaling to unity gain at fs/2, and a ready-to-run MATLAB hp_synth implementation that reproduces MATLAB's butter results.
Design IIR Band-Reject Filters
This post walks through designing IIR Butterworth band-reject filters and provides two MATLAB synthesis functions, br_synth1.m and br_synth2.m. br_synth1 accepts a null frequency plus an upper -3 dB frequency, while br_synth2 takes lower and upper -3 dB frequencies. The author demonstrates an example where a 2nd-order prototype yields a 4th-order H(z), prints b and a coefficients, and plots the response using freqz.
Design IIR Bandpass Filters
Designing Butterworth IIR bandpass filters is easier than it looks when you start from a lowpass prototype. This post walks through the s-domain lowpass-to-bandpass transform, bilinear digital mapping, and the bp_synth.m Matlab implementation that produces scaled numerator and denominator coefficients. Practical pole-zero intuition and Matlab examples help you verify magnitude and group-delay behavior for real sampling rates and bandwidths.
Phase and Amplitude Calculation for a Pure Complex Tone in a DFT
Cedron Dawg derives compact, exact formulas to recover the phase and amplitude of a single complex tone from a DFT bin when the tone frequency is known. The paper turns the complex bin value into closed-form expressions using a sine-fraction amplitude correction and a simple phase shift, and includes working code plus a numeric example for direct implementation.
Feedback Controllers - Making Hardware with Firmware. Part 7. Turbo-charged DSP Oscillators
You can extract high-quality, high-sample-rate sine waves from FPGAs even when floating-point units are constrained by latency. This article compares Intel's NCO IP (multiplier option) with floating-point recursive biquads on Cyclone V and Cyclone 10 GX, and explains a boosted-sample-rate technique that pushes performance toward a 48Msps DAC target. Practical measurement results, spectral data, and resource/cost trade-offs are highlighted.
Linear Feedback Shift Registers for the Uninitiated, Part XII: Spread-Spectrum Fundamentals
Jason Sachs shows why LFSR-generated pseudonoise is a natural fit for direct-sequence spread spectrum, then walks through Fourier basics, spectral plots, and runnable Python examples. The article demonstrates how DSSS multiplies a UART bitstream with a chipping sequence to spread energy, how despreading concentrates the desired signal while scrambling narrowband interference, and how multiple transmitters can share bandwidth when using uncorrelated sequences.
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.
An Alternative Form of the Pure Real Tone DFT Bin Value Formula
Cedron Dawg derives an alternative exact formula for DFT bin values of a pure real tone, sacrificing algebraic simplicity for better numerical behavior near integer-valued frequencies. By rewriting cosine differences as products of sines and shifting to a delta frame of reference, the derivation avoids catastrophic cancellation and preserves precision for near-integer tones. The analysis also shows the integer-frequency case is a degenerate limit that yields the familiar M/2 e^{iφ} bin value.
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.
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.
Discrete Wavelet Transform Filter Bank Implementation (part 1)
David Valencia walks through a practical implementation of discrete wavelet transform filter banks, focusing on cascading branches and efficient equivalent filters. He contrasts DWT and DFT resolution behavior and shows how cascading the low-pass branch sharpens frequency division while the high-pass path remains unchanged. Code pointers and a preview of formfilters() demonstrate how to compute only the needed samples by combining filters with upsampling.
Sensors Expo - Trip Report & My Best Video Yet!
Stephane Boucher turns a first-time Sensors Expo visit into a fun travelogue and a polished conference highlights video. He mixes candid trip anecdotes from Moncton to San Jose, electric-scooter discoveries, Santa Cruz detours, Airbnb tips, and on-the-floor expo footage. The post culminates in what he calls his best highlights reel yet, plus a follow-up video focused on embedded and IoT.
The DFT of Finite-Length Time-Reversed Sequences
Rick Lyons digs into a surprisingly under-documented corner of DSP, showing how finite-length time reversal changes a sequence's DFT. The post distinguishes flip and circular time-reversal, gives closed-form DFT relationships, and explains why modulo N arithmetic matters. Engineers get ready-to-use tables and derivations that clarify when and how time reversal affects spectral analysis.
DSP Algorithm Implementation: A Comprehensive Approach
This post lays out a practical pathway for taking DSP algorithms from high level simulation to production hardware, comparing GPP, DSP, FPGA and ASIC platforms. It presents a stepwise methodology starting with nested loop programs, then exposing parallelism with data flow graphs, using SystemC transaction level modeling to bridge to Verilog or VHDL, and explains why that flow speeds design and simulation.
Linear Feedback Shift Registers for the Uninitiated, Part XV: Error Detection and Correction
CRCs and Hamming codes look a lot less magical when you view them as redundancy with a purpose. Jason Sachs walks from parity bits and checksums into finite-field polynomial arithmetic, then shows how CRCs map cleanly onto LFSRs and how Hamming codes use syndromes to locate single-bit errors. It is a practical tour of error detection and correction, with enough worked examples to make the theory feel usable.
Learn About Transmission Lines Using a Discrete-Time Model
A simple discrete-time approach makes lossless transmission-line behavior easy to simulate and visualize. The post introduces MATLAB functions tline and wave_movie to model uniform lossless lines with resistive terminations, compute time and frequency responses, and animate travelling waves. A microstrip pulse example shows how reflections produce ringing and how source matching nearly eliminates it, making this a practical learning tool.
Number Theory for Codes
If CRCs have felt like black magic, this post peels back the curtain with basic number theory and polynomial arithmetic over GF(2). It shows how fixed-width processor arithmetic becomes arithmetic in a finite field, how bit sequences are treated as polynomials, and why primitive polynomials generate every nonzero element. You also get practical insights on CRC implementation with byte tables and LFSRs.
How to Find a Fast Floating-Point atan2 Approximation
This post shows how a compact, fast atan2 can be built from a Remez-derived arctangent approximation and a matching 3rd-order polynomial. It walks through using Boost's remez_minimax to recover coefficients 0.97239411 and -0.19194795, integrating the polynomial into an atan2 with quadrant reduction, and applying branch reduction, bit tricks, and SSE2 SIMD to cut runtime while keeping max error under about 0.005 radians.
Embedded World 2018 - The Interviews
Stephane Boucher brought video gear to Embedded World 2018 and teamed up with Jacob Beningo to capture concise vendor interviews that focus on real product news. The videos showcase Percepio's new Tracealyzer with a drone demo, Intrinsic ID's method for creating device-unique IDs from manufacturing variations, and SEGGER's broader toolset including embOS now certified by TÜV SÜD. Watch for short demos and expert explanations.
Embedded World 2018 - More Videos!
Two cinematic videos from Embedded World 2018 turn the show floor into slow-motion, stabilized footage using a Zhiyun Crane gimbal and a Sony a6300. One is a SEGGER booth highlights piece featuring Rolf Segger and Axel Wolf, the other is a roaming montage with appearances from Jacob Beningo, Micheal Barr, and Alan Hawse. Stephane asks viewers to enable audio and share feedback.
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.
A Recipe for a Common Logarithm Table
Cedron Dawg shows how to construct a base-10 logarithm table from scratch using only pencil-and-paper math. The recipe combines simple series for e and ln(1+x) with clever factoring and neighbor-based recurrences so minimal square-root work is required. Along the way the post explains a practical algorithm, high-accuracy interpolation and inverse-log reconstruction so you can reproduce published log tables by hand.
Benford's law solved with DSP
Steve Smith shows that standard DSP tools give a clean, intuitive explanation of Benford's law by treating leading-digit counts as signals on the number line and using convolution and Fourier analysis. He publishes the full derivation as an online chapter after traditional journals showed little interest. The result highlights how time- and spatial-domain DSP techniques can be applied to numeric distributions.
A Narrow Bandpass Filter in Octave or Matlab
Building very narrow FIR bandpass filters at high sample rates often yields extremely long impulse responses. This post shows a practical Octave/Matlab implementation that uses complex downconversion to baseband plus a multistage Matrix IFIR and running-sum cascade to slash computation. With the provided example (48 kHz, 850 Hz center, 10 Hz passband) you get <1 dB ripple and >60 dB stopband while running 20x to 100x faster than a single-stage FIR.
Back from ESC Boston
Stephane nearly skipped ESC Boston, but going turned into a productive mix of networking, informal meetups, and on-the-floor filming. He captures candid encounters with speakers and vendors, learns how small shows differ from larger expos, and outlines practical follow-ups like booth highlight videos and speaker hospitality suggestions. The post is an encouraging read for engineers weighing the value of regional conferences and DIY event coverage.
Linear Feedback Shift Registers for the Uninitiated, Part XIII: System Identification
Jason Sachs shows how the output of a linear feedback shift register can be used for active system identification, not just spread-spectrum testing. The article compares traditional sine-wave probing with LFSR-based PRBS methods, demonstrates a worked Ra-Rb-C example, and unpacks practical issues such as reflected pseudonoise, ADC quantization, sample counts, and noise-shaping tricks to improve estimates.
SEGGER's 25th Anniversary Video
Stephane Boucher spent a week at SEGGER's headquarters and distilled that visit into a tight, two-minute 25th anniversary video. The post highlights rising production value, thanks to softbox lighting and a two-camera setup that allows seamless wide-to-tight cuts and emotional close-ups. Stephane invites readers to watch full screen, leave feedback and thumbs-up on YouTube, and suggests future coverage like product launches or companies with happy engineers.
Do Multirate Systems Have Transfer Functions?
Multirate systems can fool you into thinking standard z-domain analysis always applies. Rick Lyons shows why CIC decimation and Hogenauer implementations do not have a single z-domain transfer function from the input to the downsampled output, because downsampling breaks the one-to-one frequency mapping of LTI systems. Use the cascaded-subfilter H(z) up to the decimation point, then explicitly account for aliasing when predicting the decimated spectrum.
