Digital Filter Instructions from IKEA?
This is a wordless example of a folded FIR filter. Swedish “Bygglek” = build and play.
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.
Add a Power Marker to a Power Spectral Density (PSD) Plot
Read absolute power directly from a PSD plot with a simple MATLAB helper. The author presents psd_mkr, a function that computes the PSD with pwelch and overlays a power marker in three modes: normal for narrowband tones, band-power for integrated power over a specified bandwidth, and 1 Hz for noise density readings. Examples show how bin summing, window loss, and scalloping are handled for accurate measurements.
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.
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.
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.
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.
Second Order Discrete-Time System Demonstration
Want a hands-on way to see how continuous second-order dynamics appear in discrete time? Neil Robertson converts a canonical H(s) to H(z), shows z-plane pole mapping for different damping ratios, and walks through impulse-invariance scaling and zero placement. The post includes a MATLAB function so_demo.m that computes numerator and denominator coefficients, plots poles, and compares impulse and frequency responses so you can experiment with sampling effects.
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.
Fractional Delay FIR Filters
You can realize arbitrary fractional-sample delays with standard FIR filters by shifting a sinc impulse response and removing symmetry, then windowing the result. This post shows a practical window-method implementation using Chebyshev windows, gives Matlab functions (frac_delay_fir.m and frac_delay_lpf.m) in the appendix, and walks through examples that demonstrate the delay, magnitude trade-offs, and how increasing taps widens the flat-delay bandwidth.
Simple but Effective Spectrum Averaging
In this article, I provide a Matlab function that performs exponential PSD averaging, using first-order infinite impulse response (IIR) filtering to continuously average the PSD bins. This approach works well for computing the spectrum of a long-duration signal over time, because the spectrum is constantly updated as new PSD’s are computed. Conveniently, the time constant of the PSD averaging is determined by the single adjustable parameter α. I also provide a Matlab function for conventional (unweighted) PSD averaging. Neither function requires any canned code other than the Fast Fourier Transform (FFT), although I do use the Matlab hann window function for convenience.
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.
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.
Add the Hilbert Transformer to Your DSP Toolkit, Part 2
This post shows a simple practical route to a Hilbert transformer by starting from a half-band FIR filter and tweaking its symmetry. It walks through a 19-tap example synthesized with Matlab's firpm (Parks-McClellan), explains the required frequency scaling, and shows how even-numbered taps become (or can be forced) zero through symmetry and coefficient quantization. Useful design rules are summarized for choosing ntaps.
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.
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.
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.
Multiplierless Half-band Filters and Hilbert Transformers
This article provides coefficients of multiplierless Finite Impulse Response 7-tap, 11-tap, and 15-tap half-band filters and Hilbert Transformers. Since Hilbert transformer coefficients are simply related to half-band coefficients, multiplierless Hilbert transformers are easily derived from multiplierless half-bands.
The Power Spectrum
You can get absolute power from a DFT, not just relative spectra. In this post Neil Robertson shows how to convert FFT outputs into watts per bin using Parseval's theorem, how to form one-sided spectra, and how to normalize windows so power is preserved. Matlab examples demonstrate bin-centered and between-bin sinusoids, leakage, scalloping, and how to recover component power by summing bins.
Second Order Discrete-Time System Demonstration
Want a hands-on way to see how continuous second-order dynamics appear in discrete time? Neil Robertson converts a canonical H(s) to H(z), shows z-plane pole mapping for different damping ratios, and walks through impulse-invariance scaling and zero placement. The post includes a MATLAB function so_demo.m that computes numerator and denominator coefficients, plots poles, and compares impulse and frequency responses so you can experiment with sampling effects.
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.
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.
The Discrete Fourier Transform and the Need for Window Functions
The FFT alone can mislead: capturing a finite-length signal with a rectangular window smears energy across frequency, producing spectral leakage that hides real components. This post explains the origin of leakage, shows how tapered windows such as the Hanning window suppress sidelobes, and demonstrates the tradeoff between sidelobe suppression and mainlobe widening while covering practical tips on zero-padding and record length.
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.
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.
Compute the Frequency Response of a Multistage Decimator
This post shows a practical way to compute the full frequency response of a multistage decimator by representing every stage at the input sample rate. The author walks through upsampling lower-rate FIR coefficients, convolving to form the overall impulse response, and taking a DFT, then demonstrates how aliasing and stopband placement affect the aliased components. Example Matlab code and plots illustrate each step.
Canonic Signed Digit (CSD) Representation of Integers
Canonic Signed Digit (CSD) encoding slashes the number of nonzero bits in integer coefficients, enabling multiplierless FIR filters implemented with shifts and adds. This post uses MATLAB code to demonstrate CSD rules, show how negative values work, and plot the distribution of signed digits as bit width changes. It finishes with practical techniques to minimize signed digits per coefficient for area and power efficient filter designs.
Compute Modulation Error Ratio (MER) for QAM
Neil Robertson shows how to define and compute Modulation Error Ratio (MER) for QAM using a simplified baseband model and decision-slice errors. The post derives per-symbol and averaged MER formulas, explains when MER tracks carrier-to-noise ratio under AWGN and matched root-Nyquist filters, and provides example Pav values for QAM-16 and QAM-64 plus a Matlab script and practical tips.
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.
The Discrete Fourier Transform as a Frequency Response
Neil Robertson shows that the discrete frequency response H(k) of an FIR filter is exactly the DFT of its impulse response h(n). He derives the continuous H(ω) and discrete H(k) using complex exponentials for a four-tap FIR, then replaces h(n) with x(n) to recover the general DFT formula. The post keeps the math simple and calls out topics left for separate treatment, such as windowing and phase.
