Smaller DFTs from bigger DFTs

Aditya Dua January 22, 20198 comments

Let's consider the following hypothetical situation: You have a sequence $x$ with $N/2$ points and a black box which can compute the DFT (Discrete Fourier Transform) of an $N$ point sequence. How will you use the black box to compute the $N/2$ point DFT of $x$? While the problem may appear to be a bit contrived, the answer(s) shed light on some basic yet insightful and useful properties of the DFT.

On a related note, the reverse problem of computing an $N$...

A Brief Introduction To Romberg Integration

Rick Lyons January 16, 201911 comments

This blog briefly describes a remarkable integration algorithm, called "Romberg integration." The algorithm is used in the field of numerical analysis but it's not so well-known in the world of DSP.

To show the power of Romberg integration, and to convince you to continue reading, consider the notion of estimating the area under the continuous x(t) = sin(t) curve based on the five x(n) samples represented by the dots in Figure 1.

The results of performing a Trapezoidal Rule, a...

Use Matlab Function pwelch to Find Power Spectral Density – or Do It Yourself

Neil Robertson January 13, 201928 comments

In my last post, we saw that finding the spectrum of a signal requires several steps beyond computing the discrete Fourier transform (DFT)[1].  These include windowing the signal, taking the magnitude-squared of the DFT, and computing the vector of frequencies.  The Matlab function pwelch [2] performs all these steps, and it also has the option to use DFT averaging to compute the so-called Welch power spectral density estimate [3,4].

In this article, I’ll present some...

Microprocessor Family Tree

Rick Lyons January 10, 20195 comments

Below is a little microprocessor history. Perhaps some of the ol' timers here will recognize a few of these integrated circuits. I have a special place in my heart for the Intel 8080 chip.

Image copied, without permission, from the now defunct Creative Computing magazine, Vol. 11, No. 6, June 1985.

A Markov View of the Phase Vocoder Part 2

Christian Yost January 8, 2019

Last post we motivated the idea of viewing the classic phase vocoder as a Markov process. This was due to the fact that the input signal’s features are unknown to the computer, and the phase advancement for the next synthesis frame is entirely dependent on the phase advancement of the current frame. We will dive a bit deeper into this idea, and flesh out some details which we left untouched last week. This includes the effect our discrete Fourier transform has on the...

A Markov View of the Phase Vocoder Part 1

Christian Yost January 8, 2019

Hello! This is my first post on I have a blog that I run on my website, In order to engage with the larger DSP community, I'd like to occasionally post my more engineering heavy writing here and get your thoughts.

Today we will look at the phase vocoder from a different angle by bringing some probability into the discussion. This is the first part in a short series. Future posts will expand further upon the ideas...

Evaluate Window Functions for the Discrete Fourier Transform

Neil Robertson December 18, 2018

The Discrete Fourier Transform (DFT) operates on a finite length time sequence to compute its spectrum.  For a continuous signal like a sinewave, you need to capture a segment of the signal in order to perform the DFT.  Usually, you also need to apply a window function to the captured signal before taking the DFT [1 - 3].  There are many different window functions and each produces a different approximation of the spectrum.  In this post, we’ll present Matlab code that...

Feedback Controllers - Making Hardware with Firmware. Part 10. DSP/FPGAs Behaving Irrationally

Steve Maslen November 22, 2018

This article will look at a design approach for feedback controllers featuring  low-latency "irrational" characteristics to enable the creation of physical components such as transmission lines. Some thought will also be given as to the capabilities of the currently utilized Intel Cyclone V, the new Cyclone 10 GX and the upcoming Xilinx Versal floating-point FPGAs/ACAPs.    

Fig 1. Making a Transmission Line, with the Circuit Emulator



Polar Coding Notes: A Simple Proof

Lyons Zhang November 8, 2018

For any B-DMC $W$, the channels $\{W_N^{(i)}\}$ polarize in the sense that, for any fixed $\delta \in (0, 1)$, as $N$ goes to infinity through powers of two, the fraction of indices $i \in \{1, \dots, N\}$ for which $I(W_N^{(i)}) \in (1 − \delta, 1]$ goes to $I(W)$ and the fraction for which $I(W_N^{(i)}) \in [0, \delta)$ goes to $1−I(W)^{[1]}$.

Mrs. Gerber’s Lemma

Mrs. Gerber’s Lemma provides a lower bound on the entropy of the modulo-$2$ sum of two binary random...

Computing Large DFTs Using Small FFTs

Rick Lyons June 23, 200821 comments

It is possible to compute N-point discrete Fourier transforms (DFTs) using radix-2 fast Fourier transforms (FFTs) whose sizes are less than N. For example, let's say the largest size FFT software routine you have available is a 1024-point FFT. With the following trick you can combine the results of multiple 1024-point FFTs to compute DFTs whose sizes are greater than 1024.

The simplest form of this idea is computing an N-point DFT using two N/2-point FFT operations. Here's how the trick...

How Discrete Signal Interpolation Improves D/A Conversion

Rick Lyons May 28, 20121 comment
This blog post is also available in pdf format. Download here.

Earlier this year, for the Linear Audio magazine, published in the Netherlands whose subscribers are technically-skilled hi-fi audio enthusiasts, I wrote an article on the fundamentals of interpolation as it's used to improve the performance of analog-to-digital conversion. Perhaps that article will be of some value to the subscribers of Here's what I wrote:

We encounter the process of digital-to-analog...

The Number 9, Not So Magic After All

Rick Lyons October 1, 20146 comments

This blog is not about signal processing. Rather, it discusses an interesting topic in number theory, the magic of the number 9. As such, this blog is for people who are charmed by the behavior and properties of numbers.

For decades I've thought the number 9 had tricky, almost magical, qualities. Many people feel the same way. I have a book on number theory, whose chapter 8 is titled "Digits — and the Magic of 9", that discusses all sorts of interesting mathematical characteristics of the...

TCP/IP interface (Matlab/Octave)

Markus Nentwig June 17, 201210 comments

Communicate with measurement instruments via Ethernet (no-toolbox-Matlab or Octave)


Measurement automation is digital signal processing in a wider sense: Getting a digital signal from an analog world usually involves some measurement instruments, for example a spectrum analyzer. Modern instruments, and also many off-the-shelf prototyping boards such as FPGA cards [1] or microcontrollers [2] are able to communicate via Ethernet. Here, I provide some basic mex-functions (compiled C...

A Beginner's Guide To Cascaded Integrator-Comb (CIC) Filters

Rick Lyons March 26, 202013 comments

This blog discusses the behavior, mathematics, and implementation of cascaded integrator-comb filters.

Cascaded integrator-comb (CIC) digital filters are computationally-efficient implementations of narrowband lowpass filters, and are often embedded in hardware implementations of decimation, interpolation, and delta-sigma converter filtering.

After describing a few applications of CIC filters, this blog introduces their structure and behavior, presents the frequency-domain...

A poor man's Simulink

Markus Nentwig January 24, 20153 comments

Glue between Octave and NGSPICE for discrete- and continuous time cosimulation (download) Keywords: Octave, SPICE, Simulink


Many DSP problems have close ties with the analog world. For example, a switched-mode audio power amplifier uses a digital control loop to open and close power transistors driving an analog filter. There are commercial tools for digital-analog cosimulation: Simulink comes to mind, and mainstream EDA vendors support VHDL-AMS or Verilog-A in their...

Sinusoidal Frequency Estimation Based on Time-Domain Samples

Rick Lyons April 20, 201719 comments

The topic of estimating a noise-free real or complex sinusoid's frequency, based on fast Fourier transform (FFT) samples, has been presented in recent blogs here on For completeness, it's worth knowing that simple frequency estimation algorithms exist that do not require FFTs to be performed . Below I present three frequency estimation algorithms that use time-domain samples, and illustrate a very important principle regarding so called "exact"...

Spectral Flipping Around Signal Center Frequency

Rick Lyons November 7, 20074 comments

Most of us are familiar with the process of flipping the spectrum (spectral inversion) of a real signal by multiplying that signal's time samples by (-1)n. In that process the center of spectral rotation is fs/4, where fs is the signal's sample rate in Hz. In this blog we discuss a different kind of spectral flipping process.

Consider the situation where we need to flip the X(f) spectrum in Figure 1(a) to obtain the desired Y(f) spectrum shown in Figure 1(b). Notice that the center of...

Signal Processing Contest in Python (PREVIEW): The Worst Encoder in the World

Jason Sachs September 7, 20136 comments

When I posted an article on estimating velocity from a position encoder, I got a number of responses. A few of them were of the form "Well, it's an interesting article, but at slow speeds why can't you just take the time between the encoder edges, and then...." My point was that there are lots of people out there which take this approach, and don't take into account that the time between encoder edges varies due to manufacturing errors in the encoder. For some reason this is a hard concept...

Back from Embedded World 2019 - Funny Stories and Live-Streaming Woes

Stephane Boucher March 1, 20191 comment

When the idea of live-streaming parts of Embedded World came to me,  I got so excited that I knew I had to make it happen.  I perceived the opportunity as a win-win-win-win.  

  • win #1 - Engineers who could not make it to Embedded World would be able to sample the huge event, 
  • win #2 - The organisation behind EW would benefit from the extra exposure
  • win #3 - Lecturers and vendors who would be live-streamed would reach a (much) larger audience
  • win #4 - I would get...