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.
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.
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.
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.
Add the Hilbert Transformer to Your DSP Toolkit, Part 1
Learn how the Hilbert transformer creates a 90-degree phase-shifted quadrature component without down-conversion, and why it is simply a special FIR filter. Part 1 defines the transformer, derives its ideal frequency response H(ω)=j for ω<0 and -j for ω≥0, and walks through Matlab examples that demonstrate phase shifting and image attenuation for bandpass signals.
Book Recommendation "What is Mathematics?"
Richard Courant and Herbert Robbins' What is Mathematics? is a lucid, classic survey that still rewards engineers who want a concept-first refresher. The author praises the book's calculus chapters as concise and readable, recommending specific sections on complex numbers, functions and limits, and calculus for practical study. Note that linear algebra is not covered and the 1996 edition adds a short Ian Stewart chapter on recent developments.
Evaluate Noise Performance of Discrete-Time Differentiators
Differentiators can be wildly different at rejecting noise, even when they share the same usable bandwidth. Neil Robertson introduces the Differentiator Noise Power Ratio, a practical Gaussian-noise metric and a compact formula that uses the filter coefficients to quantify output noise and SNR loss. The post also gives MATLAB guidance for designing and comparing FIR differentiators so you can pick or build filters with much better noise 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.
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.
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.
Interpolation Basics
Neil Robertson demonstrates interpolation by an integer factor using a frequency-domain approach, showing how zero-insertion followed by an FIR low-pass filter reconstructs a higher-rate signal. The article walks through spectra, passband and stopband selection, and a 41-tap Parks-McClellan filter example applied to a Chebyshev-window test signal. Matlab code and percent-error plots are included so engineers can reproduce and evaluate the method.
Learn to Use the Discrete Fourier Transform
Discrete-time sequences arise in many ways: a sequence could be a signal captured by an analog-to-digital converter; a series of measurements; a signal generated by a digital modulator; or simply the coefficients of a digital filter. We may wish to know the frequency spectrum of any of these sequences. The most-used tool to accomplish this is the Discrete Fourier Transform (DFT), which computes the discrete frequency spectrum of a discrete-time sequence. The DFT is easily calculated using software, but applying it successfully can be challenging. This article provides Matlab examples of some techniques you can use to obtain useful DFT’s.
Use Matlab Function pwelch to Find Power Spectral Density -- or Do It Yourself
Neil Robertson walks through using Matlab's pwelch and shows how to implement PSD estimation yourself with fft. The post uses concrete examples and complete m-files to demonstrate window selection, converting pxx (W/Hz) to W/bin, Welch DFT averaging, and a worked C/N0 calculation. Readers get practical, runnable recipes for accurate spectrum units, variance reduction with averaging, and peak-power extraction.
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.
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.
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.
Simplest Calculation of Half-band Filter Coefficients
Half-band FIR filters put the cutoff at one-quarter of the sampling rate, and nearly half their coefficients are exactly zero, which makes them highly efficient for decimation-by-2 and interpolation-by-2. This post shows the straightforward window-method derivation of half-band coefficients from the ideal sinc impulse response, providing a clear, hands-on explanation for engineers learning filter design. It also points to equiripple options such as Matlab's firhalfband and a later Parks-McClellan implementation.
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.
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.
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.
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.
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.
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.
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.
IIR Bandpass Filters Using Cascaded Biquads
This post provides a Matlab function that builds Butterworth bandpass IIR filters by cascading second-order biquad sections. The biquad approach, implemented in Direct Form II, reduces sensitivity to coefficient quantization, which matters most for narrowband filters. The included biquad_bp function computes each section's feedforward and feedback coefficients plus gains from a lowpass prototype order, center frequency, bandwidth, and sampling rate.
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.
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 First-Order IIR Filter -- More than Meets the Eye
While we might be inclined to disdain the simple first-order infinite impulse response (IIR) filter, it is not so simple that we can’t learn something from it. Studying it can teach DSP math skills, and it is a very useful filter in its own right. In this article, we’ll examine the time response of the filter, compare the first-order IIR filter to the FIR moving average filter, use it to smooth a noisy signal, compute the functional form of the impulse response, and find the frequency response.
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.
