## Stereophonic Amplitude-Panning: A Derivation of the 'Tangent Law'

In a recent Forum post here on dsprelated.com the audio signal processing subject of stereophonic amplitude-panning was discussed. And in that Forum thread the so-called "Tangent Law", the fundamental principle of stereophonic amplitude-panning, was discussed. However, none of the Forum thread participants had ever seen a derivation of the Tangent Law. This blog presents such a derivation and if this topic interests you, then please read on.

The notion of stereophonic amplitude-panning is...

## Live Streaming from Embedded World!

For those of you who won't be attending Embedded World this year, I will try to be your eyes and ears by video streaming live from the show floor.

I am not talking improvised streaming from a phone, but real, high quality HD streaming with a high-end camera and a device that will bond three internet connections (one wifi and two cellular) to ensure a steady, and hopefully reliable, stream. All this to hopefully give those of you who cannot be there in person a virtual...

## The Phase Vocoder Transform

1 IntroductionI would like to look at the phase vocoder in a fairly ``abstract'' way today. The purpose of this is to discuss a method for measuring the quality of various phase vocoder algorithms, and building off a proposed measure used in [2]. There will be a bit of time spent in the domain of continuous mathematics, thus defining a phase vocoder function or map rather than an algorithm. We will be using geometric visualizations when possible while pointing out certain group theory...

## Compute the Frequency Response of a Multistage Decimator

Figure 1a shows the block diagram of a decimation-by-8 filter, consisting of a low-pass finite impulse response (FIR) filter followed by downsampling by 8 [1]. A more efficient version is shown in Figure 1b, which uses three cascaded decimate-by-two filters. This implementation has the advantages that only FIR 1 is sampled at the highest sample rate, and the total number of filter taps is lower.

The frequency response of the single-stage decimator before downsampling is just...

## Smaller DFTs from bigger DFTs

IntroductionLet'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

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

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

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

IntroductionLast 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...

## 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...

## Simple Discrete-Time Modeling of Lossy LC Filters

There are many software applications that allow modeling LC filters in the frequency domain. But sometimes it is useful to have a time domain model, such as when you need to analyze a mixed analog and DSP system. For example, the system in Figure 1 includes an LC filter as well as a DSP portion. The LC filter could be an anti-alias filter, a channel filter, or some other LC network. For a design using undersampling, the filter would be bandpass [1]. By modeling...

## Linear Feedback Shift Registers for the Uninitiated, Part XV: Error Detection and Correction

Last time, we talked about Gold codes, a specially-constructed set of pseudorandom bit sequences (PRBS) with low mutual cross-correlation, which are used in many spread-spectrum communications systems, including the Global Positioning System.

This time we are wading into the field of error detection and correction, in particular CRCs and Hamming codes.

Ernie, You Have a Banana in Your EarI have had a really really tough time writing this article. I like the...

## Using Mason's Rule to Analyze DSP Networks

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...

## Spectral Flipping Around Signal Center Frequency

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...

## The DFT of Finite-Length Time-Reversed Sequences

Recently I've been reading papers on underwater acoustic communications systems and this caused me to investigate the frequency-domain effects of time-reversal of time-domain sequences. I created this blog because there is so little coverage of this topic in the literature of DSP.

This blog reviews the two types of time-reversal of finite-length sequences and summarizes their discrete Fourier transform (DFT) frequency-domain characteristics.The Two Types of Time-Reversal in DSP

...## Canonic Signed Digit (CSD) Representation of Integers

In my last post I presented Matlab code to synthesize multiplierless FIR filters using Canonic Signed Digit (CSD) coefficients. I included a function dec2csd1.m (repeated here in Appendix A) to convert decimal integers to binary CSD values. Here I want to use that function to illustrate a few properties of CSD numbers.

In a binary signed-digit number system, we allow each binary digit to have one of the three values {0, 1, -1}. Thus, for example, the binary value 1 1...

## Shared-multiplier polyphase FIR filter

Keywords: FPGA, interpolating decimating FIR filter, sample rate conversion, shared multiplexed pipelined multiplier

Discussion, working code (parametrized Verilog) and Matlab reference design for a FIR polyphase resampler with arbitrary interpolation and decimation ratio, mapped to one multiplier and RAM.

IntroductionA polyphase filter can be as straightforward as multirate DSP ever gets, if it doesn't turn into a semi-deterministic, three-legged little dance between input, output and...

## The History of CIC Filters: The Untold Story

If you have ever studied or designed a cascaded integrator-comb (CIC) lowpass filter then surely you've read Eugene Hogenauer's seminal 1981 IEEE paper where he first introduced the CIC filter to the signal processing world [1]. As it turns out, Hogenauer's famous paper was not the first formal document describing and proposing CIC filters. Here's the story.

In the Fall of 1979 Eugene Hogenauer was finalizing his development of the CIC filter, the filter now used in so many multirate signal...

## Summary of ROC Rules

This is a very short guide on how to find all possible outcomes of a system where Region of Convergence (ROC) and the original signal is not known.

## Ten Little Algorithms, Part 6: Green’s Theorem and Swept-Area Detection

Other articles in this series:

- Part 1: Russian Peasant Multiplication
- Part 2: The Single-Pole Low-Pass Filter
- Part 3: Welford's Method (And Friends)
- Part 4: Topological Sort
- Part 5: Quadratic Extremum Interpolation and Chandrupatla's Method

This article is mainly an excuse to scribble down some cryptic-looking mathematics — Don’t panic! Close your eyes and scroll down if you feel nauseous — and...

## Sampling bandpass signals

Sampling bandpass signals 1.1 IntroductionIt is known [1], [3] that bandpass signals can be sampled with a sampling frequency which is lower than the sampling frequency according to the sampling theorem.

Fig. 1 shows an example of how the spectrum of a bandpass signal sampled with $f_s$ (Fig. 1a) arises in the baseband with $−f_s / 2 ≤ f < f_s/2$. The bandpass signal is assumed to have a center frequency $f_c = (f_{max} + f_{min})/2$ and bandwidth $\Delta f...

## Spread the Word and Run a Chance to Win a Bundle of Goodies from Embedded World

Do you have a Twitter and/or Linkedin account?

If you do, please consider paying close attention for the next few days to the EmbeddedRelated Twitter account and to my personal Linkedin account (feel free to connect). This is where I will be posting lots of updates about how the EmbeddedRelated.tv live streaming experience is going at Embedded World.

The most successful this live broadcasting experience will be, the better the chances that I will be able to do it...

## Using the DFT as a Filter: Correcting a Misconception

I have read, in some of the literature of DSP, that when the discrete Fourier transform (DFT) is used as a filter the process of performing a DFT causes an input signal's spectrum to be frequency translated down to zero Hz (DC). I can understand why someone might say that, but I challenge that statement as being incorrect. Here are my thoughts.

Using the DFT as a Filter It may seem strange to think of the DFT as being used as a filter but there are a number of applications where this is...

## Design a DAC sinx/x Corrector

This post provides a Matlab function that designs linear-phase FIR sinx/x correctors. It includes a table of fixed-point sinx/x corrector coefficients for different DAC frequency ranges.

A sinx/x corrector is a digital (or analog) filter used to compensate for the sinx/x roll-off inherent in the digital to analog conversion process. In DSP math, we treat the digital signal applied to the DAC is a sequence of impulses. These are converted by the DAC into contiguous pulses...

## The DFT Output and Its Dimensions

The Discrete Fourier Transform, or DFT, converts a signal from discrete time to discrete frequency. It is commonly implemented as and used as the Fast Fourier Transform (FFT). This article will attempt to clarify the format of the DFT output and how it is produced.

Living in the real world, we deal with real signals. The data we typically sample does not have an imaginary component. For example, the voltage sampled by a receiver is a real value at a particular point in time. Let’s...

## Waveforms that are their own Fourier Transform

Mea Culpa

There are many scary things about writing a technical book. Can I make the concepts clear? It is worth the effort? Will it sell? But all of these pale compared to the biggest fear: What if I'm just plain wrong? Not being able to help someone is one thing, but leading them astray is far worse.

My book on DSP has now been published for almost ten years. I've found lots of typos, a few misstatements, and many places where the explanations confuse even me. But I have been lucky;...

## Understanding Radio Frequency Distortion

OverviewThe topic of this article are the effects of radio frequency distortions on a baseband signal, and how to model them at baseband. Typical applications are use as a simulation model or in digital predistortion algorithms.

IntroductionTransmitting and receiving wireless signals usually involves analog radio frequency circuits, such as power amplifiers in a transmitter or low-noise amplifiers in a receiver.Signal distortion in those circuits deteriorates the link quality. When...

## Python scipy.signal IIR Filter Design Cont.

In the previous post the Python scipy.signal iirdesign function was disected. We reviewed the basics of filter specification and reviewed how to use the iirdesign function to design IIR filters. The previous post I only demonstrated low pass filter designs. The following are examples how to use the iirdesign function for highpass, bandpass, and stopband filters designs.

Highpass FilterThe following is a highpass filter design for the different filter...

## Shared-multiplier polyphase FIR filter

Keywords: FPGA, interpolating decimating FIR filter, sample rate conversion, shared multiplexed pipelined multiplier

Discussion, working code (parametrized Verilog) and Matlab reference design for a FIR polyphase resampler with arbitrary interpolation and decimation ratio, mapped to one multiplier and RAM.

IntroductionA polyphase filter can be as straightforward as multirate DSP ever gets, if it doesn't turn into a semi-deterministic, three-legged little dance between input, output and...