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Modeling a Continuous-Time System with Matlab

Neil Robertson June 6, 20172 comments

Many of us are familiar with modeling a continuous-time system in the frequency domain using its transfer function H(s) or H(jω).  However, finding the time response can be challenging, and traditionally involves finding the inverse Laplace transform of H(s).  An alternative way to get both time and frequency responses is to transform H(s) to a discrete-time system H(z) using the impulse-invariant transform [1,2].  This method provides an exact match to the continuous-time...


How to Find a Fast Floating-Point atan2 Approximation

Nic Taylor May 26, 201715 comments
Context Over a short period of time, I came across nearly identical approximations of the two parameter arctangent function, atan2, developed by different companies, in different countries, and even in different decades. Fascinated with how the coefficients used in these approximations were derived, I set out to find them. This atan2 implementation is based around a rational approximation of arctangent on the domain -1 to 1:

$$ atan(z) \approx \dfrac{z}{1.0 +...


Exact Near Instantaneous Frequency Formulas Best at Peaks (Part 1)

Cedron Dawg May 12, 2017
Introduction

This is an article that is a another digression from trying to give a better understanding of the Discrete Fourier Transform (DFT). Although it is not as far off as the last blog article.

A new family of formulas for calculating the frequency of a single pure tone in a short interval in the time domain is presented. They are a generalization of Equation (1) from Rick Lyons' recent blog article titled "Sinusoidal Frequency Estimation Based on Time-Domain Samples"[1]. ...


A Recipe for a Common Logarithm Table

Cedron Dawg April 29, 2017
Introduction

This 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

Rick Lyons April 20, 201719 comments

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

Cedron Dawg April 13, 20171 comment
Introduction

This 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

Cedron Dawg March 20, 20179 comments
Introduction

This 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

Cedron Dawg 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

Neil Robertson 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

Qasim Chaudhari January 25, 201717 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...


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

David Norwood 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.


Model Signal Impairments at Complex Baseband

Neil Robertson December 11, 20195 comments

In this article, we develop complex-baseband models for several signal impairments: interfering carrier, multipath, phase noise, and Gaussian noise.  To provide concrete examples, we’ll apply the impairments to a QAM system. The impairment models are Matlab functions that each use at most seven lines of code.  Although our example system is QAM, the models can be used for any complex-baseband signal.

I used a very simple complex-baseband model of a QAM system in my last


The Discrete Fourier Transform as a Frequency Response

Neil Robertson February 4, 20236 comments

The discrete frequency response H(k) of a Finite Impulse Response (FIR) filter is the Discrete Fourier Transform (DFT) of its impulse response h(n) [1].  So, if we can find H(k) by whatever method, it should be identical to the DFT of h(n).  In this article, we’ll find H(k) by using complex exponentials, and we’ll see that it is indeed identical to the DFT of h(n).

Consider the four-tap FIR filter in Figure 1, where each block labeled Ts represents a delay of one...


Exponential Smoothing with a Wrinkle

Cedron Dawg December 17, 20154 comments
Introduction

This is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by providing a set of preprocessing filters to improve the resolution of the DFT. Because of the exponential nature of sinusoidal functions, they have special mathematical properties when exponential smoothing is applied to them. These properties are derived and explained in this blog article.

Basic Exponential Smoothing

Exponential smoothing is also known as...


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

Qasim Chaudhari 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...


Third-Order Distortion of a Digitally-Modulated Signal

Neil Robertson June 9, 2020
Analog designers are always harping about amplifier third-order distortion.  Why?  In this article, we’ll look at why third-order distortion is important, and simulate a QAM signal with third-order distortion.

In the following analysis, we assume that signal phase at the amplifier output is not a function of amplitude.  With this assumption, the output y of a non-ideal amplifier can be written as a power series of the input signal x:

$$y=...


Modeling Anti-Alias Filters

Neil Robertson September 26, 2021

Digitizing a signal using an Analog to Digital Converter (ADC) usually requires an anti-alias filter, as shown in Figure 1a.  In this post, we’ll develop models of lowpass Butterworth and Chebyshev anti-alias filters, and compute the time domain and frequency domain output of the ADC for an example input signal.  We’ll also model aliasing of Gaussian noise.  I hope the examples make the textbook explanations of aliasing seem a little more real.  Of course, modeling of...


Coefficients of Cascaded Discrete-Time Systems

Neil Robertson March 4, 2018

In this article, we’ll show how to compute the coefficients that result when you cascade discrete-time systems.  With the coefficients in hand, it’s then easy to compute the time or frequency response.  The computation presented here can also be used to find coefficients of mixed discrete-time and continuous-time systems, by using a discrete time model of the continuous-time portion [1].

This article is available in PDF format for...


Setting Carrier to Noise Ratio in Simulations

Neil Robertson April 11, 2021

When simulating digital receivers, we often want to check performance with added Gaussian noise.  In this article, I’ll derive the simple equations for the rms noise level needed to produce a desired carrier to noise ratio (CNR or C/N).  I also provide a short Matlab function to generate a noise vector of the desired level for a given signal vector.

Definition of C/N

The Carrier to noise ratio is defined as the ratio of average signal power to noise power for a modulated...


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

Cedron Dawg 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...