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.
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.
ADC Clock Jitter Model, Part 2 – Random Jitter
Neil Robertson shows how to simulate ADC sample-clock random jitter in Matlab, moving from band-limited Gaussian noise to wideband and close-in phase noise. The post highlights practical artifacts such as aliasing of wideband clock noise, the 20*log10 dependence of jitter sidebands on input frequency, and why cubic interpolation plus a custom noise_filter produces accurate rms and spectral results engineers can trust.
ADC Clock Jitter Model, Part 1 -- Deterministic Jitter
Clock jitter on ADC sample clocks corrupts high-frequency signals, and this post builds a practical MATLAB model to show exactly how deterministic (periodic) jitter maps into phase modulation and discrete sidebands. The author explains a parabolic-interpolation approach using twice-rate samples, demonstrates examples from single tones to pulses, and matches simulation spectra to closed-form sideband formulas so engineers can predict jitter effects.
Phase or Frequency Shifter Using a Hilbert Transformer
A Hilbert transformer converts a real input into an analytic I+jQ pair, enabling phase shifts and frequency shifts while keeping real inputs and outputs. This article shows Matlab implementations (31-tap FIR with Hamming or Blackman windows), derives y = I cosθ - Q sinθ for phase and frequency shifting, and highlights practical limits from finite taps and coefficient/NCO quantization.
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.
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.
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.
Interpolator Design: Get the Stopbands Right
In this article, I present a simple approach for designing interpolators that takes the guesswork out of determining the stopbands.
Matlab Code to Synthesize Multiplierless FIR Filters
Learn how to build multiplierless FIR lowpass filters in Matlab using Canonic Signed-Digit coefficients. The post explains converting Parks-McClellan floating-point taps to scaled integers, then to exact CSD digits, and includes two m-files that search maintap scaling to minimize signed digits while preserving the filter response. Practical notes cover external gain compensation, the 2/3 full-scale CSD limit, and sensitivity to pass/stop edges.
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.
Plotting Discrete-Time Signals
Neil Robertson demonstrates a practical interpolate-by-8 FIR approach to make sampled signals look like their continuous-time counterparts when plotted. The post explains a 121-tap filter designed for signals up to 0.4*fs, shows Matlab examples for a sinusoid and a filtered pulse, and highlights the transient and design trade-offs so you can reproduce clean plots with the supplied interp_by_8.m code.
A Simplified Matlab Function for Power Spectral Density
Neil Robertson provides a tiny Matlab wrapper around pwelch that simplifies PSD computation by preselecting a Kaiser window, default overlap, and converting units from W/Hz to dBW/bin. Call psd_simple(x,nfft,fs) to get PdB and a frequency vector, with nfft controlling whether DFT averaging is used. The post includes examples showing the effect of averaging and explains the Kaiser window processing loss.
Peak to Average Power Ratio and CCDF
Setting digital modulator levels depends on peak-to-average power ratio, because random signals produce occasional high peaks that cause clipping. This post shows how to compute the CCDF of PAPR from a signal vector, with MATLAB code and examples for a sine wave and Gaussian noise. The examples reveal the fixed 3.01 dB PAPR of a sine and the need for large sample counts to capture rare AWGN peaks.
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.
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.
Find Aliased ADC or DAC Harmonics (with animation)
If a sinewave drives an ADC or DAC, device nonlinearities create harmonics that can fold back as aliases above Nyquist. This post shows a simple Matlab model, using an NCO, a static nonlinearity, and a DFT to generate spectra and reveal aliased harmonics, with animated illustrations to make aliasing intuitive. The approach works for both ADC and DAC measurement setups and highlights realistic effects like quantization noise.
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.
Interpolator Design: Get the Stopbands Right
In this article, I present a simple approach for designing interpolators that takes the guesswork out of determining the stopbands.
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.
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.
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.
A Direct Digital Synthesizer with Arbitrary Modulus
Need exact sampled tones on a coarse grid without a huge sine table? This post shows how to build a Direct Digital Synthesizer with an arbitrary modulus so the output frequency is exactly k·fs/L, using a look-up table as small as 20 entries for the 10 MHz/0.5 MHz-step example. It also explains fixed-point LUT rounding, accumulator bit sizing, and how to produce quadrature outputs when L is multiple of 4.
