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).
Deconvolution by least squares (Using the power of linear algebra in signal processing).
When we deal with our normal discrete signal processing operations, like FIR/IIR filtering, convolution, filter design, etc. we normally think of the signals as a constant stream of numbers that we put in a sequence
The Most Interesting FIR Filter Equation in the World: Why FIR Filters Can Be Linear Phase
This blog discusses a little-known filter characteristic that enables real- and complex-coefficient tapped-delay line FIR filters to exhibit linear phase behavior. That is, this blog answers the question:
What is the constraint on real- and complex-valued FIR filters that guarantee linear phase behavior in the frequency domain?I'll declare two things to convince you to continue reading.
Declaration# 1: "That the coefficients must be symmetrical" is not a correct
Phase and Amplitude Calculation for a Pure Real Tone in a DFT: Method 1
Introduction
This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving exact formulas for the phase and amplitude of a non-integer frequency real tone in a DFT. The linearity of the Fourier Transform is exploited to reframe the problem as the equivalent of finding a set of coordinates in a specific vector space. The found coordinates are then used to calculate the phase and amplitude of the pure real tone in the DFT. This article...
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...
DFT Bin Value Formulas for Pure Real 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 real tones. The formula is used to explain the well known properties of the DFT. A sample program is included, with its output, to numerically demonstrate the veracity of the formula. This article builds on the ideas developed in my previous two blog articles:
DFT Graphical Interpretation: Centroids of Weighted Roots of Unity
IntroductionThis is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by framing it in a graphical interpretation. The bin calculation formula is shown to be the equivalent of finding the center of mass, or centroid, of a set of points. Various examples are graphed to illustrate the well known properties of DFT bin values. This treatment will only consider real valued signals. Complex valued signals can be analyzed in a similar manner with...
The Exponential Nature of the Complex Unit Circle
IntroductionThis is an article to hopefully give an understanding to Euler's magnificent equation:
$$ e^{i\theta} = cos( \theta ) + i \cdot sin( \theta ) $$
This equation is usually proved using the Taylor series expansion for the given functions, but this approach fails to give an understanding to the equation and the ramification for the behavior of complex numbers. Instead an intuitive approach is taken that culminates in a graphical understanding of the equation.
Complex...Sum of Two Equal-Frequency Sinusoids
Some time ago I reviewed the manuscript of a book being considered by the IEEE Press publisher for possible publication. In that manuscript the author presented the following equation:
Being unfamiliar with Eq. (1), and being my paranoid self, I wondered if that equation is indeed correct. Not finding a stock trigonometric identity in my favorite math reference book to verify Eq. (1), I modeled both sides of the equation using software. Sure enough, Eq. (1) is not correct. So then I...
Constrained Integer Behavior
The wheels go round and round, round and round ...
Integer arithmetic is ubiquitous in digital hardware implementations, it's prolific in the control and data-paths. When using fixed width (constrained) integers, overflow and underflow is business as usual.
Building with IntegersThe subtitle of this post mentions a wheel - before I get to the wheel I want to look at an example. The recursive-windowed-averager (rwa, a.k.a moving average)...
The Zeroing Sine Family of Window Functions
IntroductionThis is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by introducing a class of well behaved window functions that the author believes to be previously unrecognized. The definition and some characteristics are displayed. The heavy math will come in later articles. This is an introduction to the family, and a very special member of it.
This is one of my longer articles. The bulk of the material is in the front half. The...
Implementing Impractical Digital Filters
This blog discusses a problematic situation that can arise when we try to implement certain digital filters. Occasionally in the literature of DSP we encounter impractical digital IIR filter block diagrams, and by impractical I mean block diagrams that cannot be implemented. This blog gives examples of impractical digital IIR filters and what can be done to make them practical.
Implementing an Impractical Filter: Example 1
Reference [1] presented the digital IIR bandpass filter...
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...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...
Hidden Linear Algebra in DSP
Linear algebra (LA) is usually thought of as a blunt theoretical subject. However, LA is found hidden in many DSP algorithms used widely in practice.
An obvious clue in finding LA in DSP is the linearity assumption used in theoretical analysis of systems for modelling or design. A standard modelling example for this case would be linear time invariant (LTI) systems. LTI are usually used to model flat wireless communication channels. LTI systems are also used in the design of digital filter...
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...
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...Feedback Controllers - Making Hardware with Firmware. Part 6. Self-Calibration Related.
This article will consider the engineering of a self-calibration & self-test capability to enable the project hardware to be configured and its basic performance evaluated and verified, ready for the development of the low-latency controller DSP firmware and closed-loop applications. Performance specifications will be documented in due course, on the project website here.
- Part 6: Self-Calibration, Measurements and Signalling (this part)
- Part 5:
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...Polar Coding Notes: Channel Combining and Channel Splitting
Channel Combining
Channel combining is a step that combines copies of a given B-DMC $W$ in a recursive manner to produce a vector channel $W_N : {\cal X}^N \to {\cal Y}^N$, where $N$ can be any power of two, $N=2^n, n\le0^{[1]}$.
The notation $u_1^N$ as shorthand for denoting a row vector $(u_1, \dots , u_N)$.
The vector channel $W_N$ is the virtual channel between the input sequence $u_1^N$ to a linear encoder and the output sequence $y^N_1$ of $N$...
Exponential Smoothing with a Wrinkle
IntroductionThis 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...
Feedback Controllers - Making Hardware with Firmware. Part 3. Sampled Data Aspects
Some Design and Simulation Considerations for Sampled-Data ControllersThis article will continue to look at some aspects of the controllers and electronics needed to create emulated physical circuits with real-world connectivity and will look at the issues that arise in sampled-data controllers compared to continuous-domain controllers. As such, is not intended as an introduction to sampled-data systems.
- Part 1: Introduction
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...
Correlation without pre-whitening is often misleading
White Lies
Correlation, as one of the first tools DSP users add to their tool box, can automate locating a known signal within a second (usually larger) signal. The expected result of a correlation is a nice sharp peak at the location of the known signal and few, if any, extraneous peaks.
A little thought will show this to be incorrect: correlating a signal with itself is only guaranteed to give a sharp peak if the signal's samples are uncorrelated --- for example if the signal is composed...
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...Feedback Controllers - Making Hardware with Firmware. Part 8. Control Loop Test-bed
This part in the series will consider the signals, measurements, analyses and configurations for testing high-speed low-latency feedback loops and their controllers. Along with basic test signals, a versatile IFFT signal generation scheme will be discussed and implemented. A simple controller under test will be constructed to demonstrate the analysis principles in preparation for the design and evaluation of specific controllers and closed-loop applications.
Additional design...Feedback Controllers - Making Hardware with Firmware. Part 6. Self-Calibration Related.
This article will consider the engineering of a self-calibration & self-test capability to enable the project hardware to be configured and its basic performance evaluated and verified, ready for the development of the low-latency controller DSP firmware and closed-loop applications. Performance specifications will be documented in due course, on the project website here.
- Part 6: Self-Calibration, Measurements and Signalling (this part)
- Part 5:
Constrained Integer Behavior
The wheels go round and round, round and round ...Integer arithmetic is ubiquitous in digital hardware implementations, it's prolific in the control and data-paths. When using fixed width (constrained) integers, overflow and underflow is business as usual.
Building with IntegersThe subtitle of this post mentions a wheel - before I get to the wheel I want to look at an example. The recursive-windowed-averager (rwa, a.k.a moving average)...
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...Least-squares magic bullets? The Moore-Penrose Pseudoinverse
Hello,
the topic of this brief article is a tool that can be applied to a variety of problems: The Moore-Penrose Pseudoinverse.While maybe not exactly a magic bullet, it gives us least-squares optimal solutions, and that is under many circumstances the best we can reasonably expect.
I'll demonstrate its use on a short example. More details can be found for example on Wikipedia, or the Matlab documentation...
