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.
Smaller DFTs from bigger DFTs
A neat DFT puzzle turns into a tour of three useful spectral tricks. Given only an N point DFT black box, the post shows how to recover the N/2 point DFT of a shorter sequence by zero padding, zero interlacing, or repeating the data. Along the way, it highlights why some methods smooth the spectrum, why others replicate it, and how these operations relate to FFT fundamentals.
Polar Coding Notes: Channel Combining and Channel Splitting
Lyons Zhang walks through the core algebra of polar coding, showing how channel combining builds the vector channel W_N from N copies of a binary-input DMC using the polar transform G_N = B_N F^{⊗n}. The notes then define channel splitting, derive the coordinate-channel transition probabilities from the chain rule, and present the recursive formulas that let you compute W_{2N}^{(2i-1)} and W_{2N}^{(2i)} from W_N^{(i)}.
Take Control of Noise with Spectral Averaging
Spectral averaging turns noisy FFT outputs into repeatable, measurable spectra by trading time for noise control. This post explains the practical difference between RMS averaging, which reduces variance without changing the noise floor, and vector averaging, which can lower the noise floor but requires phase-coherent, triggered inputs. It also shows how linear and exponential weighting affect reaction time for live displays and measurement accuracy.
How precise is my measurement?
Precision is quantifiable, not guesswork. This post walks through practical, measurement-oriented statistics you can apply to static or dynamic signals to answer the question, "How precise is my measurement?" It focuses on using multiple samples, checking distribution assumptions, and constructing confidence intervals and levels so you can trade measurement time for a desired precision.
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.
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.
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.
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.
A brief look at multipath radio channels
Markus Nentwig walks through a hands-on RF experiment that makes multipath and fading visible using a network analyzer and simple dipole antennas. He shows how reflections produce frequency-domain notches when path differences equal half wavelengths, and how doubling distance increases free-space path loss by roughly 6 dB. The post explains why narrowband signals often see flat fading while wideband links become frequency-selective, motivating OFDM and multi-tap channel models.
Sampling bandpass signals
Bandpass signals can be sampled at rates below the usual Nyquist limit, and this note shows how the band-limited spectrum appears in baseband after sampling. Using a simple example figure, it defines the center frequency fc = (fmax + fmin)/2 and bandwidth Δf = fmax - fmin, and highlights that choosing fs less than twice the signal's highest frequency violates the sampling theorem.
Python scipy.signal IIR Filtering: An Example
Christopher Felton walks through using scipy.signal IIR filters to demodulate PWM signals, using spectrum and spectrogram analysis to show what works and what does not. He demonstrates using filtfilt to avoid phase delay, compares a single narrow IIR to a very high order FIR, and shows how staged IIR filtering and multirate ideas give much better attenuation. Includes an FPGA-ready MyHDL PWM model.
Polyphase filter / Farrows interpolation
Markus Nentwig shows how polyphase filtering and the Farrow interpolator provide a practical, computation‑efficient way to realize sub-sample delays and variable resampling. He starts from the upsample-filter-decimate view, explains how polyphase decomposition reduces per-phase work, then describes how the Farrow structure fits polynomials to coefficient banks for continuous fractional-delay control. The post includes warnings about filter choices and links to code and references.
Design study: 1:64 interpolating pulse shaping FIR
Markus Nentwig presents a practical 1:64 root-raised cosine interpolator built from cascaded FIR stages that slashes computational cost. By separating pulse shaping from rate conversion, designing each interpolator to suppress only known alias bands, and equalizing the pulse shape, the design achieves just 4.69 MACs per output, roughly 12 percent of a straight polyphase implementation while meeting EVM targets.
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.
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.
Python scipy.signal IIR Filter Design
Christopher Felton walks through designing infinite impulse response filters using scipy.signal in Python, focusing on practical specs and functions rather than theoretical derivations. He explains normalized passband and stopband definitions, gpass and gstop, and shows how iirdesign and iirfilter differ. Plots compare elliptic, Chebyshev, Butterworth and Bessel responses, highlighting steep transitions versus near-linear phase tradeoffs.
Exact Near Instantaneous Frequency Formulas Best at Peaks (Part 2)
Cedron Dawg derives a second family of exact time domain formulas for single-tone frequency estimation that trade a few extra calculations for improved noise robustness. Built from [1+cos]^k binomial weighting of neighbor-pair sums, the closed-form estimators are exact and are best evaluated at signal peaks for real tones, while complex tones do not share the zero-crossing limitation. Coefficients up to k=9 are provided.
Python scipy.signal IIR Filter Design Cont.
Christopher Felton continues his practical tour of SciPy's iirdesign, moving beyond lowpass examples to show highpass, bandpass, and stopband designs with concise, code-focused explanations. He highlights how ellip and cheby2 let you tighten specifications for sharper transitions, and shows that the iirdesign workflow is consistent across filter types. Read for clear, reusable examples to produce IIR filter coefficients with scipy.signal.
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.
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.
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.
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.
Python scipy.signal IIR Filter Design Cont.
Christopher Felton continues his practical tour of SciPy's iirdesign, moving beyond lowpass examples to show highpass, bandpass, and stopband designs with concise, code-focused explanations. He highlights how ellip and cheby2 let you tighten specifications for sharper transitions, and shows that the iirdesign workflow is consistent across filter types. Read for clear, reusable examples to produce IIR filter coefficients with scipy.signal.
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.
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.
Should DSP Undergraduate Students Study z-Transform Regions of Convergence?
Rick Lyons argues z-transform regions of convergence are mostly a classroom abstraction with little practical use for real-world DSP engineers. For all stable LTI impulse responses encountered in practice the ROC includes the unit circle, so DTFT and DFT exist and ROC analysis rarely affects implementation. He notes digital oscillators are a notable exception, and suggests reallocating classroom time to more practical engineering topics.
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.
How precise is my measurement?
Precision is quantifiable, not guesswork. This post walks through practical, measurement-oriented statistics you can apply to static or dynamic signals to answer the question, "How precise is my measurement?" It focuses on using multiple samples, checking distribution assumptions, and constructing confidence intervals and levels so you can trade measurement time for a desired precision.
