DFT Bin Value Formulas for Pure Complex Tones
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
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
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
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
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
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
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
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
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 frequency and...Digital PLL's -- Part 1
1. IntroductionFigure 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...
Wavelets I - From Filter Banks to the Dilation Equation
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...
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...
Coefficients of Cascaded Discrete-Time Systems
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...
The Power Spectrum
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...
Exact Frequency Formula for a Pure Real Tone in a DFT
IntroductionThis is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving an exact formula for the frequency of a real tone in a DFT. According to current teaching, this is not possible, so this article should be considered a major theoretical advance in the discipline. The formula is presented in a few different formats. Some sample calculations are provided to give a numerical demonstration of the formula in use. This article is...
How precise is my measurement?
Some might argue that measurement is a blend of skepticism and faith. While time constraints might make you lean toward faith, some healthy engineering skepticism should bring you back to statistics. This article reviews some practical statistics that can help you satisfy one common question posed by skeptical engineers: “How precise is my measurement?” As we’ll see, by understanding how to answer it, you gain a degree of control over your measurement time.
An accurate, precise...There and Back Again: Time of Flight Ranging between Two Wireless Nodes
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...
Modeling a Continuous-Time System with Matlab
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...
Decimator Image Response
Note: this is an improved version of a post I made to the dsp forum a few weeks ago.
This article presents a way to compute and plot the image response of a decimator. I’m defining the image response as the unwanted spectrum of the impulse response after downsampling, relative to the desired passband response.
Consider a decimate-by-4 filter with fs= 4 Hz, to which we apply the signal spectrum shown in Figure 1. The desired signal is the CW component at 0.22 Hz,...
Phase and Amplitude Calculation for a Pure Complex Tone in a DFT
IntroductionThis 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...
Modeling a Continuous-Time System with Matlab
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...
Multilayer Perceptrons and Event Classification with data from CODEC using Scilab and Weka
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...
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...
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...
There and Back Again: Time of Flight Ranging between Two Wireless Nodes
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...
Filter a Rectangular Pulse with no Ringing
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...Design IIR Band-Reject Filters
In this post, I show how to design IIR Butterworth band-reject filters, and provide two Matlab functions for band-reject filter synthesis. Earlier posts covered IIR Butterworth lowpass [1] and bandpass [2] filters. Here, the function br_synth1.m designs band-reject filters based on null frequency and upper -3 dB frequency, while br_synth2.m designs them based on lower and upper -3 dB frequencies. I’ll discuss the differences between the two approaches later in this...
Digital PLL’s, Part 3 – Phase Lock an NCO to an External Clock
Sometimes you may need to phase-lock a numerically controlled oscillator (NCO) to an external clock that is not related to the system clocks of your ASIC or FPGA. This situation is shown in Figure 1. Assuming your system has an analog-to-digital converter (ADC) available, you can sync to the external clock using the scheme shown in Figure 2. This time-domain PLL model is similar to the one presented in Part 1 of this series on digital PLL’s [1]. In that PLL, we...
Decimator Image Response
Note: this is an improved version of a post I made to the dsp forum a few weeks ago.
This article presents a way to compute and plot the image response of a decimator. I’m defining the image response as the unwanted spectrum of the impulse response after downsampling, relative to the desired passband response.
Consider a decimate-by-4 filter with fs= 4 Hz, to which we apply the signal spectrum shown in Figure 1. The desired signal is the CW component at 0.22 Hz,...
Approximating the area of a chirp by fitting a polynomial
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...
