## Generating Partially Correlated Random Variables

IntroductionIt is often useful to be able to generate two or more signals with specific cross-correlations. Or, more generally, we would like to specify an $\left(N \times N\right)$ covariance matrix, $\mathbf{R}_{xx}$, and generate $N$ signals which will produce this covariance matrix.There are many applications in which this technique is useful. I discovered a version of this method while analysing radar systems, but the same approach can be used in a very wide range of...

## Angle Addition Formulas from Euler's Formula

IntroductionThis 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 CoordinatesThe more common way of...

## Smaller DFTs from bigger DFTs

IntroductionLet's consider the following hypothetical situation: You have a sequence $x$ with $N/2$ points and a black box which can compute the DFT (Discrete Fourier Transform) of an $N$ point sequence. How will you use the black box to compute the $N/2$ point DFT of $x$? While the problem may appear to be a bit contrived, the answer(s) shed light on some basic yet insightful and useful properties of the DFT.

On a related note, the reverse problem of computing an $N$...

## 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 $N$...

## Take Control of Noise with Spectral Averaging

Most engineers have seen the moment-to-moment fluctuations that are common with instantaneous measurements of a supposedly steady spectrum. You can see these fluctuations in magnitude and phase for each frequency bin of your spectrogram. Although major variations are certainly reason for concern, recall that we don’t live in an ideal, noise-free world. After verifying the integrity of your measurement setup by checking connections, sensors, wiring, and the like, you might conclude that the...

## 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...## 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...## Phase and Amplitude Calculation for a Pure Complex Tone in a DFT using Multiple Bins

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 several DFT bin values and knowing the frequency. This article is functionally an extension of my prior article "Phase and Amplitude Calculation for a Pure Complex Tone in a DFT"[1] which used only one bin for a complex tone, but it is actually much more similar to my approach for real...

## 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...

## Feedback Controllers - Making Hardware with Firmware. Part 7. Turbo-charged DSP Oscillators

This article will look at some DSP Sine-wave oscillators and will show how an FPGA with limited floating-point performance due to latency, can be persuaded to produce much higher sample-rate sine-waves of high quality.Comparisons will be made between implementations on Intel Cyclone V and Cyclone 10 GX FPGAs. An Intel numerically controlled oscillator

## 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 snippetThis 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...

## Smaller DFTs from bigger DFTs

IntroductionLet's consider the following hypothetical situation: You have a sequence $x$ with $N/2$ points and a black box which can compute the DFT (Discrete Fourier Transform) of an $N$ point sequence. How will you use the black box to compute the $N/2$ point DFT of $x$? While the problem may appear to be a bit contrived, the answer(s) shed light on some basic yet insightful and useful properties of the DFT.

On a related note, the reverse problem of computing an $N$...

## 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...

## 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...

## 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'...

## 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:

## 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...

## 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...

## 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 snippetThis 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,...

## Pulse Shaping in Single-Carrier Communication Systems

Some common conceptual hurdles for beginning communications engineers have to do with "Pulse Shaping" or the closely-related, even synonymous, topics of "matched filtering", "Nyquist filtering", "Nyquist pulse", "pulse filtering", "spectral shaping", etc. Some of the confusion comes from the use of terms like "matched filter" which has a broader meaning in the more general field of signal processing or detection theory. Likewise "Raised Cosine" has a different meaning or application in this...

## 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...

## Frequency Dependence in Free Space Propagation

Introduction

It seems to be fairly common knowledge, even among practicing professionals, that the efficiency of propagation of wireless signals is frequency dependent. Generally it is believed that lower frequencies are desirable since pathloss effects will be less than they would be at higher frequencies. As evidence of this, the Friis Transmission Equation[i] is often cited, the general form of which is usually written as:

Pr = Pt Gt Gr ( λ / 4πd )2 (1)

where the...