## A Recipe for a Common Logarithm Table

IntroductionThis is an article that is a digression from trying to give a better understanding to the Discrete Fourier Transform (DFT).

A method for building a table of Base 10 Logarithms, also known as Common Logarithms, is featured using math that can be done with paper and pencil. The reader is assumed to have some familiarity with logarithm functions. This material has no dependency on the material in my previous blog articles.

If you were ever curious about how...

## Sinusoidal Frequency Estimation Based on Time-Domain Samples

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 dsprelated.com. 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"...

## Three Bin Exact Frequency Formulas for a Pure Complex Tone in a DFT

IntroductionThis is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by deriving exact formulas for the frequency of a complex tone in a DFT. This time it is three bin versions. Although the problem is similar to the two bin version in my previous blog article "A Two Bin Exact Frequency Formula for a Pure Complex Tone in a DFT"[1], a slightly different approach is taken using linear algebra concepts. Because of an extra degree of freedom...

## A Two Bin Exact Frequency Formula for a Pure Complex Tone in a DFT

IntroductionThis is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by deriving an exact formula for the frequency of a complex tone in a DFT. It is basically a parallel treatment to the real case given in Exact Frequency Formula for a Pure Real Tone in a DFT. Since a real signal is the sum of two complex signals, the frequency formula for a single complex tone signal is a lot less complicated than for the real case.

Theoretical...## DFT Bin Value Formulas for Pure Complex Tones

IntroductionThis is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by deriving an analytical formula for the DFT of pure complex tones and an alternative variation. It is basically a parallel treatment to the real case given in DFT Bin Value Formulas for Pure Real Tones. In order to understand how a multiple tone signal acts in a DFT it is necessary to first understand how a single pure tone acts. Since a DFT is a linear transform, the...

## New Video: Parametric Oscillations

I just posted this last night. It's kinda off-topic from the mission of the channel, but I realized that it had been months since I'd posted a video, and having an excuse to build on helped keep me on track.

## Some Thoughts on Sampling

Some time ago, I came across an interesting problem. In the explanation of sampling process, a representation of impulse sampling shown in Figure 1 below is illustrated in almost every textbook on DSP and communications. The question is: how is it possible that during sampling, the frequency axis gets scaled by $1/T_s$ -- a very large number? For an ADC operating at 10 MHz for example, the amplitude of the desired spectrum and spectral replicas is $10^7$! I thought that there must be...

## Fibonacci trick

I'm working on a video, tying the Fibonacci sequence into the general subject of difference equations.

Here's a fun trick: take any two consecutive numbers in the Fibonacci sequence, say 34 and 55. Now negate one and use them as the seed for the Fibonacci sequence, larger magnitude first, i.e.

$-55, 34, \cdots$

Carry it out, and you'll eventually get the Fibonacci sequence, or it's negative:

$-55, 34, -21, 13, -8, 5, -3, 2, -1, 1, 0, 1, 1 \cdots$

This is NOT a general property of difference...

## An s-Plane to z-Plane Mapping Example

While surfing around the Internet recently I encountered the 's-plane to z-plane mapping' diagram shown in Figure 1. At first I thought the diagram was neat because it's a good example of the old English idiom: "A picture is worth a thousand words." However, as I continued to look at Figure 1 I began to detect what I believe are errors in the diagram.

Reader, please take a few moments to see if you detect any errors in Figure 1.

...## 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 ...

## Pentagon Construction Using Complex Numbers

A method for constructing a pentagon using a straight edge and a compass is deduced from the complex values of the Fifth Roots of Unity. Analytic values for the points are also derived.

## A Recipe for a Common Logarithm Table

IntroductionThis is an article that is a digression from trying to give a better understanding to the Discrete Fourier Transform (DFT).

A method for building a table of Base 10 Logarithms, also known as Common Logarithms, is featured using math that can be done with paper and pencil. The reader is assumed to have some familiarity with logarithm functions. This material has no dependency on the material in my previous blog articles.

If you were ever curious about how...

## FIR sideways (interpolator polyphase decomposition)

An efficient implementation of a symmetric-FIR polyphase 1:3 interpolator that doesn't follow the usual tapped delay line-paradigm. The example exploits the impulse response symmetry and avoids four multiplications out of 10. keywords: symmetric polyphase FIR filter implementation ASIC Matlab / Octave implementation

IntroductionAn interpolating FIR filter can be implemented with a single tapped delay line, possibly going forwards and backwards for a symmetric impulse response. To...

## Design of an anti-aliasing filter for a DAC

Overview- Octaveforge / Matlab design script. Download: here
- weighted numerical optimization of Laplace-domain transfer function
- linear-phase design, optimizes vector error (magnitude and phase)
- design process calculates and corrects group delay internally
- includes sinc() response of the sample-and-hold stage in the ADC
- optionally includes multiplierless FIR filter

Digital-to-analog conversion connects digital...

## Sinusoidal Frequency Estimation Based on Time-Domain Samples

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 dsprelated.com. 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"...

## 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...

## Amplitude modulation and the sampling theorem

I am working on the 11th and probably final chapter of Think DSP, which follows material my colleague Siddhartan Govindasamy developed for a class at Olin College. He introduces amplitude modulation as a clever way to sneak up on the Nyquist–Shannon sampling theorem.

Most of the code for the chapter is done: you can check it out in this IPython notebook. I haven't written the text yet, but I'll outline it here, and paste in the key figures.

Convolution...

## A Two Bin Solution

IntroductionThis is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by showing an implementation of how the parameters of a real pure tone can be calculated from just two DFT bin values. The equations from previous articles are used in tandem to first calculate the frequency, and then calculate the amplitude and phase of the tone. The approach works best when the tone is between the two DFT bins in terms of frequency.

The Coding...## A Recipe for a Basic Trigonometry Table

IntroductionThis is an article that is give a better understanding to the Discrete Fourier Transform (DFT) by showing how to build a Sine and Cosine table from scratch. Along the way a recursive method is developed as a tone generator for a pure tone complex signal with an amplitude of one. Then a simpler multiplicative one. Each with drift correction factors. By setting the initial values to zero and one degrees and letting it run to build 45 values, the entire set of values needed...

## Instantaneous Frequency Measurement

I would like to talk about the oft used method of measuring the carrier frequency in the world of Signal Collection and Characterization world. It is an elegant technique because of its simplicity. But, of course, with simplicity, there come drawbacks (sometimes...especially with this one!).

In the world of Radar detection and characterization, one of the key characteristics of interest is the carrier frequency of the signal. If the radar is pulsed, you will have a very wide bandwidth, a...

## Discrete Wavelet Transform Filter Bank Implementation (part 2)

Following the previous blog entry: http://www.dsprelated.com/showarticle/115.php

Difference between DWT and DWPTBefore getting to the equivalent filter obtention, I first want to talk about the difference between DWT(Discrete Wavelet Transform) and DWPT (Discrete Wavelet Packet Transform). The latter is used mostly for image processing.

While DWT has a single "high-pass" branch that filters the signal with the h1 filter, the DWPT separates branches symmetricaly: this means that one...

## A Fast Real-Time Trapezoidal Rule Integrator

This blog presents a computationally-efficient network for computing real‑time discrete integration using the Trapezoidal Rule.

Background

While studying what is called "N-sample Romberg integration" I noticed that such an integration process requires the computation of many individual smaller‑sized integrations using the Trapezoidal Rule integration method [1]. My goal was to create a computationally‑fast real‑time Trapezoidal Rule integration network to increase the processing...

## Feedback Controllers - Making Hardware with Firmware. Part I. Introduction

Introduction to the topicThis is the 1st in a series of articles looking at how we can use DSP and Feedback Control Sciences along with some mixed-signal electronics and number-crunching capability (e.g. FPGA), to create arbitrary (within reason) Electrical/Electronic Circuits with real-world connectivity. Of equal importance will be the evaluation of the functionality and performance of a practical design made from modestly-priced state of the art devices.

- Part 1:

## Resolving 'Can't initialize target CPU' on TI C6000 DSPs - Part 2

Configuration

The previous article discussed CCS configuration. The prerequisite for the following discussion is a valid CCS configuration file. All references will be for CCS 3.3, but they may be used or adapted to other versions of CCS. From the previous discussion, we know that the configuration file is located at 'C:\CCStudio_v3.3\cc\bin\brddat\ccBrd0.dat'.

XDS510 Emulators

Initial discussion will address only XDS510 class emulators that support TI drivers and utilities. This will...

## Resolving 'Can't initialize target CPU' on TI C6000 DSPs - Part 1

Introduction

Today I am going to discuss some of the basics that can help prevent errors that frustrate some users. The information is directed toward TI C6000 family DSPs, but much of it also applies to other TI DSPs. In many cases they represent the user's first involvement with using Code Composer Studio [CCS] and a target board. It has been my experience that the primary cause of the "Can't initialize target CPU" error message and similar messages like "Error connecting to...

## FIR sideways (interpolator polyphase decomposition)

An efficient implementation of a symmetric-FIR polyphase 1:3 interpolator that doesn't follow the usual tapped delay line-paradigm. The example exploits the impulse response symmetry and avoids four multiplications out of 10. keywords: symmetric polyphase FIR filter implementation ASIC Matlab / Octave implementation

IntroductionAn interpolating FIR filter can be implemented with a single tapped delay line, possibly going forwards and backwards for a symmetric impulse response. To...

## Generating Partially Correlated Random Variables

IntroductionIt is often useful to be able to generate two or more signals with specific cross-correlations. Or, more generally, we would like to specify an $\left(N \times N\right)$ covariance matrix, $\mathbf{R}_{xx}$, and generate $N$ signals which will produce this covariance matrix.There are many applications in which this technique is useful. I discovered a version of this method while analysing radar systems, but the same approach can be used in a very wide range of...

## Simple Concepts Explained: Fixed-Point

IntroductionMost signal processing intensive applications on FPGA are still implemented relying on integer or fixed-point arithmetic. It is not easy to find the key ideas on quantization, fixed-point and integer arithmetic. In a series of articles, I aim to clarify some concepts and add examples on how things are done in real life. The ideas covered are the result of my professional experience and hands-on projects.

In this article I will present the most fundamental question you...

## Feedback Controllers - Making Hardware with Firmware. Part 7. Turbo-charged DSP Oscillators

This article will look at some DSP Sine-wave oscillators and will show how an FPGA with limited floating-point performance due to latency, can be persuaded to produce much higher sample-rate sine-waves of high quality.Comparisons will be made between implementations on Intel Cyclone V and Cyclone 10 GX FPGAs. An Intel numerically controlled oscillator

## Some Thoughts on Sampling

Some time ago, I came across an interesting problem. In the explanation of sampling process, a representation of impulse sampling shown in Figure 1 below is illustrated in almost every textbook on DSP and communications. The question is: how is it possible that during sampling, the frequency axis gets scaled by $1/T_s$ -- a very large number? For an ADC operating at 10 MHz for example, the amplitude of the desired spectrum and spectral replicas is $10^7$! I thought that there must be...