Learn to Use the Discrete Fourier Transform
Discrete-time sequences arise in many ways: a sequence could be a signal captured by an analog-to-digital converter; a series of measurements; a signal generated by a digital modulator; or simply the coefficients of a digital filter. We may wish to know the frequency spectrum of any of these sequences. The most-used tool to accomplish this is the Discrete Fourier Transform (DFT), which computes the discrete frequency spectrum of a discrete-time sequence. The DFT is easily calculated using software, but applying it successfully can be challenging. This article provides Matlab examples of some techniques you can use to obtain useful DFT’s.
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.
Algebra's Laws of Powers and Roots: Handle With Care
Recently, for entertainment, I tried to solve a puzzling algebra problem featured on YouTube [1]. In due course I learned that algebra’s $$(a^x)^y=a^{xy}\qquad\qquad\qquad\qquad\qquad(1)$$
Law of Powers identity is not always valid (not always true) if variable a is real and exponents x and y are complex-valued.
The fact that Eq. (1) can’t reliably be used with complex x and y exponents surprised me. And then I thought, “Humm, …what other of algebra’s identities may also...
Simple Concepts Explained: Fixed-Point
Introduction
Most 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...
Candan's Tweaks of Jacobsen's Frequency Approximation
IntroductionThis is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by explaining how a tweak to a well known frequency approximation formula makes it better, and another tweak makes it exact. The first tweak is shown to be the first of a pattern and a novel approximation formula is made from the second. It only requires a few extra calculations beyond the original approximation to come up with an approximation suitable for most...
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...
Filtering Noise: The Basics (Part 1)
Introduction
Finding signals in the presence of noise is one of the fundamental quests of the discipline of signal processing. Noise is inherently random by nature, so a probability oriented approach is needed to develop a mathematical framework for filtering (i.e. removing/suppressing) noise. This framework or discipline, formally referred to as stochastic signal processing, is often taught in graduate level engineering programs and is covered from different perspectives in excellent...
Determination of the transfer function of passive networks with MATLAB Functions
With MATLAB functions, the transfer function of passive networks can be determined relatively easily. The method is explained using the example of a passive low-pass filter of the sixth order, which is shown in Fig.1
Fig.1 Passive low-pass filter of the sixth order
If one tried, as would be logical, to calculate the transfer function starting from the input, it would be quite complicated. On the other hand, if you start from the output, the determination of this function is simple...
Sampling bandpass signals
Sampling bandpass signals 1.1 IntroductionIt is known [1], [3] that bandpass signals can be sampled with a sampling frequency which is lower than the sampling frequency according to the sampling theorem.
Fig. 1 shows an example of how the spectrum of a bandpass signal sampled with $f_s$ (Fig. 1a) arises in the baseband with $−f_s / 2 ≤ f < f_s/2$. The bandpass signal is assumed to have a center frequency $f_c = (f_{max} + f_{min})/2$ and bandwidth $\Delta f...
Simulink-Simulation of SSB demodulation
≥≥≥ Simulink-Simulation of SSB demodulation or modulation from the article “Understanding the ‘Phasing Method’ of Single Sideband Demodulation” by Richard Lyons Josef HoffmannThe article “Understanding the ‘Phasing Method’ of Single Sideband Demodulation” by Richard Lyons is a very good description of this topic. The block representation from the figures are clear and easy to understand. They are predestined for a simulation in Simulink. The simulation can help...
A Quadrature Signals Tutorial: Complex, But Not Complicated
Introduction Quadrature signals are based on the notion of complex numbers and perhaps no other topic causes more heartache for newcomers to DSP than these numbers and their strange terminology of j operator, complex, imaginary, real, and orthogonal. If you're a little unsure of the physical meaning of complex numbers and the j = √-1 operator, don't feel bad because you're in good company. Why even Karl Gauss, one the world's greatest mathematicians, called the j-operator the "shadow of...
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...
A Fixed-Point Introduction by Example
Introduction
The finite-word representation of fractional numbers is known as fixed-point. Fixed-point is an interpretation of a 2's compliment number usually signed but not limited to sign representation. It extends our finite-word length from a finite set of integers to a finite set of rational real numbers [1]. A fixed-point representation of a number consists of integer and fractional components. The bit length is defined...
Understanding the 'Phasing Method' of Single Sideband Demodulation
There are four ways to demodulate a transmitted single sideband (SSB) signal. Those four methods are:
- synchronous detection,
- phasing method,
- Weaver method, and
- filtering method.
Here we review synchronous detection in preparation for explaining, in detail, how the phasing method works. This blog contains lots of preliminary information, so if you're already familiar with SSB signals you might want to scroll down to the 'SSB DEMODULATION BY SYNCHRONOUS DETECTION'...
A Beginner's Guide to OFDM
In the recent past, high data rate wireless communications is often considered synonymous to an Orthogonal Frequency Division Multiplexing (OFDM) system. OFDM is a special case of multi-carrier communication as opposed to a conventional single-carrier system.
The concepts on which OFDM is based are so simple that almost everyone in the wireless community is a technical expert in this subject. However, I have always felt an absence of a really simple guide on how OFDM works which can...
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...Algebra's Laws of Powers and Roots: Handle With Care
Recently, for entertainment, I tried to solve a puzzling algebra problem featured on YouTube [1]. In due course I learned that algebra’s $$(a^x)^y=a^{xy}\qquad\qquad\qquad\qquad\qquad(1)$$
Law of Powers identity is not always valid (not always true) if variable a is real and exponents x and y are complex-valued.
The fact that Eq. (1) can’t reliably be used with complex x and y exponents surprised me. And then I thought, “Humm, …what other of algebra’s identities may also...
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...
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:
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.
...A Quadrature Signals Tutorial: Complex, But Not Complicated
Introduction Quadrature signals are based on the notion of complex numbers and perhaps no other topic causes more heartache for newcomers to DSP than these numbers and their strange terminology of j operator, complex, imaginary, real, and orthogonal. If you're a little unsure of the physical meaning of complex numbers and the j = √-1 operator, don't feel bad because you're in good company. Why even Karl Gauss, one the world's greatest mathematicians, called the j-operator the "shadow of...
A Fixed-Point Introduction by Example
IntroductionThe finite-word representation of fractional numbers is known as fixed-point. Fixed-point is an interpretation of a 2's compliment number usually signed but not limited to sign representation. It extends our finite-word length from a finite set of integers to a finite set of rational real numbers [1]. A fixed-point representation of a number consists of integer and fractional components. The bit length is defined...
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...
Understanding the 'Phasing Method' of Single Sideband Demodulation
There are four ways to demodulate a transmitted single sideband (SSB) signal. Those four methods are:
- synchronous detection,
- phasing method,
- Weaver method, and
- filtering method.
Here we review synchronous detection in preparation for explaining, in detail, how the phasing method works. This blog contains lots of preliminary information, so if you're already familiar with SSB signals you might want to scroll down to the 'SSB DEMODULATION BY SYNCHRONOUS DETECTION'...
A Beginner's Guide to OFDM
In the recent past, high data rate wireless communications is often considered synonymous to an Orthogonal Frequency Division Multiplexing (OFDM) system. OFDM is a special case of multi-carrier communication as opposed to a conventional single-carrier system.
The concepts on which OFDM is based are so simple that almost everyone in the wireless community is a technical expert in this subject. However, I have always felt an absence of a really simple guide on how OFDM works which can...
Python scipy.signal IIR Filtering: An Example
IntroductionIn the last posts I reviewed how to use the Python scipy.signal package to design digital infinite impulse response (IIR) filters, specifically, using the iirdesign function (IIR design I and IIR design II ). In this post I am going to conclude the IIR filter design review with an example.
Previous posts:
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...Delay estimation by FFT
Given x=sig(t) and y=ref(t), returns [c, ref(t+delta), delta)] = fitSignal(y, x);:Estimates and corrects delay and scaling factor between two signals Code snippet
This article relates to the Matlab / Octave code snippet: Delay estimation with subsample resolution It explains the algorithm and the design decisions behind it.
IntroductionThere are many DSP-related problems, where an unknown timing between two signals needs to be determined and corrected, for example, radar, sonar,...
Python scipy.signal IIR Filter Design
IntroductionThe following is an introduction on how to design an infinite impulse response (IIR) filters using the Python scipy.signal package. This post, mainly, covers how to use the scipy.signal package and is not a thorough introduction to IIR filter design. For complete coverage of IIR filter design and structure see one of the references.
Filter SpecificationBefore providing some examples lets review the specifications for a filter design. A filter...
Polyphase filter / Farrows interpolation
Hello,
this article is meant to give a quick overview over polyphase filtering and Farrows interpolation.
A good reference with more depth is for example Fred Harris' paper: http://www.signumconcepts.com/IP_center/paper018.pdf
The task is as follows: Interpolate a band-limited discrete-time signal at a variable offset between samples.In other words:Delay the signal by a given amount with sub-sample accuracy.Both mean the same.
The picture below shows samples (black) representing...
