## Four Ways to Compute an Inverse FFT Using the Forward FFT Algorithm

July 7, 20151 comment

If you need to compute inverse fast Fourier transforms (inverse FFTs) but you only have forward FFT software (or forward FFT FPGA cores) available to you, below are four ways to solve your problem.

Preliminaries To define what we're thinking about here, an N-point forward FFT and an N-point inverse FFT are described by:

$$Forward \ FFT \rightarrow X(m) = \sum_{n=0}^{N-1} x(n)e^{-j2\pi nm/N} \tag{1}$$  Inverse \ FFT \rightarrow x(n) = {1 \over N} \sum_{m=0}^{N-1}...

## Correcting an Important Goertzel Filter Misconception

Recently I was on the Signal Processing Stack Exchange web site (a question and answer site for DSP people) and I read a posted question regarding Goertzel filters [1]. One of the subscribers posted a reply to the question by pointing interested readers to a Wikipedia web page discussing Goertzel filters [2]. I noticed the Wiki web site stated that a Goertzel filter:

"...is marginally stable and vulnerable tonumerical error accumulation when computed usinglow-precision arithmetic and...

## Handy Online Simulation Tool Models Aliasing With Lowpass and Bandpass Sampling

May 4, 20151 comment

Analog Devices Inc. has posted a neat software simulation tool on their corporate web site that graphically shows the aliasing effects of both lowpass and bandpass periodic sampling. This is a nice tutorial tool for beginners in DSP.

The tool shows four important characteristics of periodic sampling:

Characteristic# 1: All input analog spectral components, regardless of their center frequencies, show up (appear) below half the sample rate in the digitized...

## Why Time-Domain Zero Stuffing Produces Multiple Frequency-Domain Spectral Images

This blog explains why, in the process of time-domain interpolation (sample rate increase), zero stuffing a time sequence with zero-valued samples produces an increased-length time sequence whose spectrum contains replications of the original time sequence's spectrum.

Background

The traditional way to interpolate (sample rate increase) an x(n) time domain sequence is shown in Figure 1.

Figure 1

The '↑ L' operation in Figure 1 means to...

## Complex Down-Conversion Amplitude Loss

This blog illustrates the signal amplitude loss inherent in a traditional complex down-conversion system. (In the literature of signal processing, complex down-conversion is also called "quadrature demodulation.")

The general idea behind complex down-conversion is shown in Figure 1(a). And the traditional hardware block diagram of a complex down-converter is shown in Figure 1(b).

Let's assume the input to our down-conversion system is an analog radio frequency (RF) signal,...

## 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)...

## The Number 9, Not So Magic After All

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...

## Sum of Two Equal-Frequency Sinusoids

Some time ago I reviewed the manuscript of a book being considered by the IEEE Press publisher for possible publication. In that manuscript the author presented the following equation:

Being unfamiliar with Eq. (1), and being my paranoid self, I wondered if that equation is indeed correct. Not finding a stock trigonometric identity in my favorite math reference book to verify Eq. (1), I modeled both sides of the equation using software. Sure enough, Eq. (1) is not correct. So then I...

## The DFT Magnitude of a Real-valued Cosine Sequence

This blog may seem a bit trivial to some readers here but, then again, it might be of some value to DSP beginners. It presents a mathematical proof of what is the magnitude of an N-point discrete Fourier transform (DFT) when the DFT's input is a real-valued sinusoidal sequence.

To be specific, if we perform an N-point DFT on N real-valued time-domain samples of a discrete cosine wave, having exactly integer k cycles over N time samples, the peak magnitude of the cosine wave's...

## Specifying the Maximum Amplifier Noise When Driving an ADC

I recently learned an interesting rule of thumb regarding the use of an amplifier to drive the input of an analog to digital converter (ADC). The rule of thumb describes how to specify the maximum allowable noise power of the amplifier [1].

The Problem Here's the situation for an ADC whose maximum analog input voltage range is –VRef to +VRef. If we drive an ADC's analog input with an sine wave whose peak amplitude is VP = VRef, the ADC's output signal to noise ratio is maximized. We'll...

## Beat Notes: An Interesting Observation

Some weeks ago a friend of mine, a long time radio engineer as well as a piano player, called and asked me,

"When I travel in a DC-9 aircraft, and I sit back near the engines, I hear this fairly loud unpleasant whump whump whump whump sound. The frequency of that sound is, maybe, two cycles per second. I think that sound is a beat frequency because the DC-9's engines are turning at a slightly different number of revolutions per second. My question is, what sort of mechanism in the airplane...

## Correcting an Important Goertzel Filter Misconception

Recently I was on the Signal Processing Stack Exchange web site (a question and answer site for DSP people) and I read a posted question regarding Goertzel filters [1]. One of the subscribers posted a reply to the question by pointing interested readers to a Wikipedia web page discussing Goertzel filters [2]. I noticed the Wiki web site stated that a Goertzel filter:

"...is marginally stable and vulnerable tonumerical error accumulation when computed usinglow-precision arithmetic and...

## Frequency Translation by Way of Lowpass FIR Filtering

Some weeks ago a question appeared on the dsp.related Forum regarding the notion of translating a signal down in frequency and lowpass filtering in a single operation [1]. It is possible to implement such a process by embedding a discrete cosine sequence's values within the coefficients of a traditional lowpass FIR filter. I first learned about this process from Reference [2]. Here's the story.

Traditional Frequency Translation Prior To Filtering

Think about the process shown in...

## Online DSP Classes: Why Such a High Dropout Rate?

Last year the IEEE Signal Processing Magazine published a lengthy article describing three university-sponsored online digital signal processing (DSP) courses [1]. The article detailed all the effort the professors expended in creating those courses and the courses' perceived values to students.

However, one fact that struck me as important, but not thoroughly addressed in the article, was the shocking dropout rate of those online courses. For two of the courses the article's...

## How Not to Reduce DFT Leakage

This blog describes a technique to reduce the effects of spectral leakage when using the discrete Fourier transform (DFT).

In late April 2012 there was a thread on the comp.dsp newsgroup discussing ways to reduce the spectral leakage problem encountered when using the DFT. One post in that thread caught my eye [1]. That post referred to a website presenting a paper describing a DFT leakage method that I'd never heard of before [2]. (Of course, not that I've heard...

## Coupled-Form 2nd-Order IIR Resonators: A Contradiction Resolved

This blog clarifies how to obtain and interpret the z-domain transfer function of the coupled-form 2nd-order IIR resonator. The coupled-form 2nd-order IIR resonator was developed to overcome a shortcoming in the standard 2nd-order IIR resonator. With that thought in mind, let's take a brief look at a standard 2nd-order IIR resonator.

Standard 2nd-Order IIR Resonator A block diagram of the standard 2nd-order IIR resonator is shown in Figure 1(a). You've probably seen that block diagram many...

## Should DSP Undergraduate Students Study z-Transform Regions of Convergence?

Not long ago I presented my 3-day DSP class to a group of engineers at Tektronix Inc. in Beaverton Oregon [1]. After I finished covering my material on IIR filters' z-plane pole locations and filter stability, one of the Tektronix engineers asked a question similar to:

"I noticed that you didn't discuss z-plane regions of      convergence here. In my undergraduate DSP class we      spent a lot of classroom and homework time on the  ...

## Why Time-Domain Zero Stuffing Produces Multiple Frequency-Domain Spectral Images

This blog explains why, in the process of time-domain interpolation (sample rate increase), zero stuffing a time sequence with zero-valued samples produces an increased-length time sequence whose spectrum contains replications of the original time sequence's spectrum.

Background

The traditional way to interpolate (sample rate increase) an x(n) time domain sequence is shown in Figure 1.

Figure 1

The '↑ L' operation in Figure 1 means to...

## The Little Fruit Market

There used to be a fruit market located at 391 San Antonio Road in Mountain View, California. In the 1990's I worked part time in Mountain View and drove past this market's building, shown in Figure 1, many times, unaware of its history. What happened at that fruit market has changed the lives of almost everyone on our planet. Here's the story.

William Shockley In 1948 the brilliant physicist William Shockley, along with John Bardeen and Walter Brattain, co-invented the transistor at Bell...