## Computing an FFT of Complex-Valued Data Using a Real-Only FFT Algorithm

February 9, 20103 comments

Someone recently asked me if I knew of a way to compute a fast Fourier transform (FFT) of complex-valued input samples using an FFT algorithm that accepts only real-valued input data. Knowing of no way to do this, I rifled through my library of hardcopy FFT articles looking for help. I found nothing useful that could be applied to this problem.

After some thinking, I believe I have a solution to this problem. Here is my idea:

Let's say our original input data is the complex-valued sequence...

## Unit Testing for Embedded Algorithms

December 21, 2009

Happy Holidays! For my premier article, I am writing about my favorite technique to use when designing and developing software- unit testing. Unit testing is a best practice when designing software. It allows the designer to verify the behavior of the software units before the entire system is complete, and it facilitates the change and growth of the software system because the developer can verify that the changes will not affect the behavior of other parts of the system. I have used...

## Using Mason's Rule to Analyze DSP Networks

August 31, 20096 comments

There have been times when I wanted to determine the z-domain transfer function of some discrete network, but my algebra skills failed me. Some time ago I learned Mason's Rule, which helped me solve my problems. If you're willing to learn the steps in using Mason's Rule, it has the power of George Foreman's right hand in solving network analysis problems.

This blog discusses a valuable analysis method (well known to our analog control system engineering brethren) to obtain the z-domain...

## The Nature of Circles

February 21, 20093 comments
What do you mean?

When calculating the mean of a list of numbers, the obvious approach is to sum them and divide by how many there are.

Suppose I give you a list of two numbers:

• 0
• 359

What is their mean? The obvious answer is 179.5.

If I told you that the numbers were compass bearings in degrees, what would your answer be then? Does 179.5 seem correct?

In the case of compass bearings, 0 is the same direction as 360. When talking about angles in the DSP world, we often talk about...

## Simultaneously Computing a Forward FFT and an Inverse FFT Using a Single FFT

January 13, 20095 comments

Most of us are familiar with the processes of using a single N-point complex FFT to: (1) perform a 2N-point FFT on real data, and (2) perform two independent N-point FFTs on real data [1–5]. In case it's of interest to someone out there, this blog gives the algorithm for simultaneously computing a forward FFT and an inverse FFT using a single radix-2 FFT.

Our algorithm is depicted by the seven steps, S1 through S7, shown in Figure 1. In that figure, we compute the x(n) inverse FFT of...

## Multiplierless Exponential Averaging

December 5, 200811 comments

This blog discusses an interesting approach to exponential averaging. To begin my story, a traditional exponential averager (also called a "leaky integrator"), shown in Figure 1(a), is commonly used to reduce noise fluctuations that contaminate relatively constant-amplitude signal measurements.

Figure 1 Exponential averaging: (a) standard network; (b) single-multiply network.

That exponential averager's difference equation is

y(n) = αx(n) + (1 –...

## Computing the Group Delay of a Filter

November 19, 200817 comments

I just learned a new method (new to me at least) for computing the group delay of digital filters. In the event this process turns out to be interesting to my readers, this blog describes the method. Let's start with a bit of algebra so that you'll know I'm not making all of this up.

Assume we have the N-sample h(n) impulse response of a digital filter, with n being our time-domain index, and that we represent the filter's discrete-time Fourier transform (DTFT), H(ω), in polar form...

## Computing Large DFTs Using Small FFTs

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...

## Linear-phase DC Removal Filter

March 30, 200826 comments

This blog describes several DC removal networks that might be of interest to the dsprelated.com readers.

Back in August 2007 there was a thread on the comp.dsp newsgroup concerning the process of removing the DC (zero Hz) component from a time-domain sequence [1]. Discussed in that thread was the notion of removing a signal's DC bias by subtracting the signal's moving average from that signal, as shown in Figure 1(a).

Figure 1.

At first I thought...

## Correlation without pre-whitening is often misleading

February 18, 20089 comments
White Lies

Correlation, as one of the first tools DSP users add to their tool box, can automate locating a known signal within a second (usually larger) signal. The expected result of a correlation is a nice sharp peak at the location of the known signal and few, if any, extraneous peaks.

A little thought will show this to be incorrect: correlating a signal with itself is only guaranteed to give a sharp peak if the signal's samples are uncorrelated --- for example if the signal is composed...

## Hidden Linear Algebra in DSP

June 17, 20105 comments

Linear algebra (LA) is usually thought of as a blunt theoretical subject. However, LA is found hidden in many DSP algorithms used widely in practice.

An obvious clue in finding LA in DSP is the linearity assumption used in theoretical analysis of systems for modelling or design. A standard modelling example for this case would be linear time invariant (LTI) systems. LTI are usually used to model flat wireless communication channels. LTI systems are also used in the design of digital filter...

## Impulse Response Approximation

August 5, 20113 comments

Recently, I stumbled upon a stepped-triangular (ST) approximation that can be implemented as a cascade of recursive running sum (RRS) filters.  The following is a short introduction to the stepped-triangular approximation.The stepped-triangular approximation was introduced by Jovanovic-Dolecek and Mitra [1] as a quantized approximation of a low-pass filter (LPF).  Figure 1 shows an example of the approximation.

[Figure 1: Stepped Approximation of a LPF...

## Correlation without pre-whitening is often misleading

February 18, 20089 comments
White Lies

Correlation, as one of the first tools DSP users add to their tool box, can automate locating a known signal within a second (usually larger) signal. The expected result of a correlation is a nice sharp peak at the location of the known signal and few, if any, extraneous peaks.

A little thought will show this to be incorrect: correlating a signal with itself is only guaranteed to give a sharp peak if the signal's samples are uncorrelated --- for example if the signal is composed...

## Simultaneously Computing a Forward FFT and an Inverse FFT Using a Single FFT

January 13, 20095 comments

Most of us are familiar with the processes of using a single N-point complex FFT to: (1) perform a 2N-point FFT on real data, and (2) perform two independent N-point FFTs on real data [1–5]. In case it's of interest to someone out there, this blog gives the algorithm for simultaneously computing a forward FFT and an inverse FFT using a single radix-2 FFT.

Our algorithm is depicted by the seven steps, S1 through S7, shown in Figure 1. In that figure, we compute the x(n) inverse FFT of...

## Least-squares magic bullets? The Moore-Penrose Pseudoinverse

October 24, 20109 comments

Hello,

the topic of this brief article is a tool that can be applied to a variety of problems: The Moore-Penrose Pseudoinverse.While maybe not exactly a magic bullet, it gives us least-squares optimal solutions, and that is under many circumstances the best we can reasonably expect.

I'll demonstrate its use on a short example. More details can be found for example on Wikipedia, or the Matlab documentation...

## There's No End to It -- Matlab Code Plots Frequency Response above the Unit Circle

October 23, 20179 comments
Reference [1] has some 3D plots of frequency response magnitude above the unit circle in the Z-plane.  I liked them enough that I wrote a Matlab function to plot the response of any digital filter this way.  I’m not sure how useful these plots are, but they’re fun to look at. The Matlab code is listed in the Appendix.

This post is available in PDF format for easy...

## A Complex Variable Detective Story – A Disconnect Between Theory and Implementation

October 14, 2014

Recently I was in the middle of a pencil-and-paper analysis of a digital 5-tap FIR filter having complex-valued coefficients and I encountered a surprising and thought-provoking problem. So that you can avoid the algebra difficulty I encountered, please read on.

A Surprising Algebra Puzzle

I wanted to derive the H(ω) equation for the frequency response of my FIR digital filter whose complex coefficients were h0, h1, h2, h3, and h4. I could then test the validity of my H(ω)...

## Exploring Human Hearing Range

October 31, 20204 comments
Human Hearing Range

In this post, I'll look at an interesting aspect of Audacity – using it to explore the threshold of human hearing. In my book Digital Signal Processing: A Gentle Introduction with Audio Examples, I go into this topic and I include a side note on the amazing hearing range of our canine companions.

Creating a Test Audio File

Audacity allows for the generation of a variety of test signals. If you click the Generate->Tone menu, it looks something like...

## Implementing a full-duplex UART using the TMS320VC33 serial port

March 16, 20112 comments

Although the TMS320VC33 serial port was designed to be used as a synchronous port, it can also be used as an asynchronous port under software control. This post describes the hardware and software needed to use a TMS320VC33 serial port as a full-duplex UART port. A schematic diagram and a lengthy code listing are provided to illustrate the solution. This note discusses the implementation of an interrupt-driven, full-duplex, asynchronous serial interface, 9600-baud UART with 8 data bits, 1...

## Unit Testing for Embedded Algorithms

December 21, 2009

Happy Holidays! For my premier article, I am writing about my favorite technique to use when designing and developing software- unit testing. Unit testing is a best practice when designing software. It allows the designer to verify the behavior of the software units before the entire system is complete, and it facilitates the change and growth of the software system because the developer can verify that the changes will not affect the behavior of other parts of the system. I have used...