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.
Design IIR Butterworth Filters Using 12 Lines of Code
Build a working lowpass IIR Butterworth filter from first principles in just 12 lines of Matlab using Neil Robertson's butter_synth.m. The post walks through the analog prototype poles, frequency pre-warping, bilinear transform pole mapping, adding N zeros at z = -1, and gain normalization so the result matches Matlab's built-in butter function. It's a compact, hands-on guide with clear formulas and code.
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.
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.
Feedback Controllers - Making Hardware with Firmware. Part 2. Ideal Model Examples
An engineer's guide to building ideal continuous-time models for hardware emulation, using TINA Spice, MATLAB and Simulink to validate controller and circuit behavior. The article shows how a passive R-C network can be emulated by an amplifier, a current measurement and a summer, with Spice, MATLAB and Simulink producing coincident Bode responses. Small phase differences between MATLAB and Simulink are noted, and sampled-data issues are slated for the next installment.
Feedback Controllers - Making Hardware with Firmware. Part I. Introduction
This first post kicks off a series on using DSP and feedback control with mixed-signal electronics and FPGAs to emulate two-terminal circuits and create low latency controllers. It frames circuit emulation as a feedback problem, highlights latency as the key practical constraint, and outlines the planned evaluation hardware, target devices, and software tools that will be used in later MATLAB/Simulink and FPGA work.
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 a DAC sinx/x Corrector
Neil Robertson provides a compact Matlab function and coefficient tables for designing linear-phase FIR sinx/x correctors to undo the DAC sinc roll-off. The post explains the sinc_corr(ntaps,fmax,fs) call, shows worked examples with ntaps=5 and different fmax values, and demonstrates fixed-point quantization including a k=512 example and CSD digit guidance. Practical notes cover corrector gain and input back-off to avoid clipping.
Bank-switched Farrow resampler
Markus Nentwig proposes a bank-switched variant of the Farrow resampler that breaks each impulse-response segment into multiple sub-segments, enabling accurate interpolation with lower-order polynomials and fewer multiplications per output. This trades increased total coefficient storage for computational savings. The post explains the concept, connects it to polyphase FIR interpolation, and provides Matlab/Octave and C example code for practical evaluation.
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.
Evaluate Window Functions for the Discrete Fourier Transform
Spectral leakage makes DFTs of continuous sinewaves misleading, and windowing is the practical workaround. This post supplies Matlab code to plot spectra of windowed sinewaves and compute figures of merit, so you can compare windows such as flattop and Chebyshev. See how sidelobe level, mainlobe bandwidth, processing loss, noise bandwidth, and scallop loss trade off to guide your window choice.
Modeling a Continuous-Time System with Matlab
Neil Robertson demonstrates a practical workflow for converting a continuous-time transfer function H(s) into an exact discrete-time H(z) using Matlab's impinvar. He walks through a 3rd-order Butterworth example, shows how to match impulse and step responses, and compares frequency response and group delay so engineers can see where the discrete model stays accurate and when sampling-rate limits cause departure.
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.
Setting Carrier to Noise Ratio in Simulations
Setting the right Gaussian noise level is easy once you know the math. This post derives simple, practical equations to compute noise density and the rms noise amplitude needed to achieve a target carrier to noise ratio at a receiver output. It shows how to get the noise-equivalent bandwidth from a discrete-time filter, how to compute N0 and sigma, and includes a MATLAB set_cnr function to generate the noise vector.
Modeling Anti-Alias Filters
Modeling anti-alias filters brings textbook aliasing examples to life. This post shows how to build discrete-time models G(z) for analog Butterworth and Chebyshev lowpass anti-alias filters, compares bilinear transform and impulse invariance, and simulates ADC input/output including aliasing of sinusoids and Gaussian noise. It concludes that impulse invariance gives better stopband accuracy and includes Matlab helper functions.
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.
Decimators Using Cascaded Multiplierless Half-band Filters
In my last post, I provided coefficients for several multiplierless half-band FIR filters. In the comment section, Rick Lyons mentioned that such filters would be useful in a multi-stage decimator. For such an application, any subsequent multipliers save on resources, since they operate at a fraction of the maximum sample frequency. We’ll examine the frequency response and aliasing of a multiplierless decimate-by-8 cascade in this article, and we’ll also discuss an interpolator cascade using the same half-band filters.
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.
Feedback Controllers - Making Hardware with Firmware. Part 2. Ideal Model Examples
An engineer's guide to building ideal continuous-time models for hardware emulation, using TINA Spice, MATLAB and Simulink to validate controller and circuit behavior. The article shows how a passive R-C network can be emulated by an amplifier, a current measurement and a summer, with Spice, MATLAB and Simulink producing coincident Bode responses. Small phase differences between MATLAB and Simulink are noted, and sampled-data issues are slated for the next installment.
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.
Modeling a Continuous-Time System with Matlab
Neil Robertson demonstrates a practical workflow for converting a continuous-time transfer function H(s) into an exact discrete-time H(z) using Matlab's impinvar. He walks through a 3rd-order Butterworth example, shows how to match impulse and step responses, and compares frequency response and group delay so engineers can see where the discrete model stays accurate and when sampling-rate limits cause departure.
Feedback Controllers - Making Hardware with Firmware. Part 9. Closing the low-latency loop
This article demonstrates combining DSP and feedback-control on an Intel Cyclone floating-point FPGA to build low-latency closed-loop circuit emulators and controllers. Using a single floating-point biquad at 1.6 Msps, an IFFT multi-tone 4.096 ms capture for wideband measurement, and MATLAB references for verification, the author achieves sub-nanosecond timing insight and applies DSP phase compensation to cancel about 100 pF of PCB parasitics.
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.
Third-Order Distortion of a Digitally-Modulated Signal
Amplifier third-order distortion is a common limiter in RF and communications chains, and Neil Robertson walks through why it matters using hands-on MATLAB simulations. He shows how a cubic nonlinearity creates IMD3 tones, causes spectral regrowth and degrades QAM constellations, and gives practical notes on estimating k3, computing ACPR from PSDs, and sampling considerations.
A Matlab Function for FIR Half-Band Filter Design
FIR Half-band filters are not difficult to design. In an earlier post [1], I showed how to design them using the window method. Here, I provide a short Matlab function halfband_synth that uses the Parks-McClellan algorithm (Matlab function firpm [2]) to synthesize half-band filters. Compared to the window method, this method uses fewer taps to achieve a given performance.
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.
