## DFT Bin Value Formulas for Pure Complex Tones

March 17, 2017
Introduction

This 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...

## Canonic Signed Digit (CSD) Representation of Integers

February 18, 2017

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...

## Minimum Shift Keying (MSK) - A Tutorial

January 25, 20179 comments

Minimum Shift Keying (MSK) is one of the most spectrally efficient modulation schemes available. Due to its constant envelope, it is resilient to non-linear distortion and was therefore chosen as the modulation technique for the GSM cell phone standard.

MSK is a special case of Continuous-Phase Frequency Shift Keying (CPFSK) which is a special case of a general class of modulation schemes known as Continuous-Phase Modulation (CPM). It is worth noting that CPM (and hence CPFSK) is a...

## New Video: Parametric Oscillations

January 4, 2017

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.

## Wavelets II - Vanishing Moments and Spectral Factorization

October 11, 2016

In the previous blog post I described the workings of the Fast Wavelet Transform (FWT) and how wavelets and filters are related. As promised, in this article we will see how to construct useful filters. Concretely, we will find a way to calculate the Daubechies filters, named after Ingrid Daubechies, who invented them and also laid much of the mathematical foundations for wavelet analysis.

Besides the content of the last post, you should be familiar with basic complex algebra, the...

## The Power Spectrum

October 8, 2016

Often, when calculating the spectrum of a sampled signal, we are interested in relative powers, and we don’t care about the absolute accuracy of the y axis.  However, when the sampled signal represents an analog signal, we sometimes need an accurate picture of the analog signal’s power in the frequency domain.  This post shows how to calculate an accurate power spectrum.

Parseval’s theorem [1,2] is a property of the Discrete Fourier Transform (DFT) that...

## Wavelets I - From Filter Banks to the Dilation Equation

September 28, 20169 comments

This is the first in what I hope will be a series of posts about wavelets, particularly about the Fast Wavelet Transform (FWT). The FWT is extremely useful in practice and also very interesting from a theoretical point of view. Of course there are already plenty of resources, but I found them tending to be either simple implementation guides that do not touch on the many interesting and sometimes crucial connections. Or they are highly mathematical and definition-heavy, for a...

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

September 24, 20166 comments

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.

...

## Digital PLL's -- Part 2

June 15, 20162 comments

In Part 1, we found the time response of a 2nd order PLL with a proportional + integral (lead-lag) loop filter.  Now let’s look at this PLL in the Z-domain [1, 2].  We will find that the response is characterized by a loop natural frequency ωn and damping coefficient ζ.

Having a Z-domain model of the DPLL will allow us to do three things:

Compute the values of loop filter proportional gain KL and integrator gain KI that give the desired loop natural...

## Digital PLL's -- Part 1

June 7, 201618 comments
1. Introduction

Figure 1.1 is a block diagram of a digital PLL (DPLL).  The purpose of the DPLL is to lock the phase of a numerically controlled oscillator (NCO) to a reference signal.  The loop includes a phase detector to compute phase error and a loop filter to set loop dynamic performance.  The output of the loop filter controls the frequency and phase of the NCO, driving the phase error to zero.

One application of the DPLL is to recover the timing in a digital...

## Demonstrating the Periodic Spectrum of a Sampled Signal Using the DFT

March 9, 201920 comments

One of the basic DSP principles states that a sampled time signal has a periodic spectrum with period equal to the sample rate.  The derivation of can be found in textbooks [1,2].  You can also demonstrate this principle numerically using the Discrete Fourier Transform (DFT).

The DFT of the sampled signal x(n) is defined as:

$$X(k)=\sum_{n=0}^{N-1}x(n)e^{-j2\pi kn/N} \qquad (1)$$

Where

X(k) = discrete frequency spectrum of time sequence x(n)

## ADC Clock Jitter Model, Part 1 – Deterministic Jitter

April 16, 20185 comments

Analog to digital converters (ADC’s) have several imperfections that affect communications signals, including thermal noise, differential nonlinearity, and sample clock jitter [1, 2].  As shown in Figure 1, the ADC has a sample/hold function that is clocked by a sample clock.  Jitter on the sample clock causes the sampling instants to vary from the ideal sample time.  This transfers the jitter from the sample clock to the input signal.

In this article, I present a Matlab...

## Multilayer Perceptrons and Event Classification with data from CODEC using Scilab and Weka

November 25, 2015

For my first blog, I thought I would introduce the reader to Scilab [1] and Weka [2].  In order to illustrate how they work, I will put together a script in Scilab that will sample using the microphone and CODEC on your PC and save the waveform as a CSV file.  Then, we can take the CSV file and open it in Weka.  Once in Weka, we have a lot of paths to consider in order to classify it.  I use the term classify loosely since there are many things you can do with data sets...

## There and Back Again: Time of Flight Ranging between Two Wireless Nodes

October 23, 20175 comments

With the growth in the Internet of Things (IoT) products, the number of applications requiring an estimate of range between two wireless nodes in indoor channels is growing very quickly as well. Therefore, localization is becoming a red hot market today and will remain so in the coming years.

One question that is perplexing is that many companies now a days are offering cm level accurate solutions using RF signals. The conventional wireless nodes usually implement synchronization...

## Plotting Discrete-Time Signals

September 15, 20195 comments

A discrete-time sinusoid can have frequency up to just shy of half the sample frequency.  But if you try to plot the sinusoid, the result is not always recognizable.  For example, if you plot a 9 Hz sinusoid sampled at 100 Hz, you get the result shown in the top of Figure 1, which looks like a sine.  But if you plot a 35 Hz sinusoid sampled at 100 Hz, you get the bottom graph, which does not look like a sine when you connect the dots.  We typically want the plot of a...

## Filter a Rectangular Pulse with no Ringing

May 12, 201610 comments

To filter a rectangular pulse without any ringing, there is only one requirement on the filter coefficients:  they must all be positive.  However, if we want the leading and trailing edge of the pulse to be symmetrical, then the coefficients must be symmetrical.  What we are describing is basically a window function.

Consider a rectangular pulse 32 samples long with fs = 1 kHz.  Here is the Matlab code to generate the pulse:

N= 64; fs= 1000; % Hz sample...

## Approximating the area of a chirp by fitting a polynomial

November 15, 20158 comments

Once in a while we need to estimate the area of a dataset in which we are interested. This area could give us, for example, force (mass vs acceleration) or electric power (electric current vs charge).

One way to do that is fitting a curve on our data, and let's face it: this is not that easy. In this post we will work on this issue using Python and its packages. If you do not have Python installed on your system, check here how to...

## Phase and Amplitude Calculation for a Pure Complex Tone in a DFT

January 6, 2018
Introduction

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving exact formulas to calculate the phase and amplitude of a pure complex tone from a DFT bin value and knowing the frequency. This is a much simpler problem to solve than the corresponding case for a pure real tone which I covered in an earlier blog article[1]. In the noiseless single tone case, these equations will be exact. In the presence of noise or other tones...

## Compute the Frequency Response of a Multistage Decimator

February 10, 20192 comments

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...

## Angle Addition Formulas from Euler's Formula

March 16, 20198 comments
Introduction

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT), but only indirectly. The main intent is to get someone who is uncomfortable with complex numbers a little more used to them and relate them back to already known Trigonometric relationships done in Real values. It is essentially a followup to my first blog article "The Exponential Nature of the Complex Unit Circle".

Polar Coordinates

The more common way of...