Questions tagged [math]

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Question about delayed sampled sinusoid math expression

I have been studying the digital audio processing by using the book <Designing Audio Effect Plugins in C++>. For analog Sinusoid: Complex Sinusoid = $e^{jωt}$ Delayed Sinusoid = $e^{jω(t−n)} = e^...
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1answer
57 views

Why Does the Median Filter Minimize the Absolute Value Error $L_1$ Cost Function?

I can easily prove that the mean filter minimizes the square error $L_2$ cost function using simple calculus. However, how do you prove that the median filter is optimal with respect the absolute ...
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1answer
42 views

mathematical expression to detect modulated data within a vector

Assume I have such modulated data which is, for example, $x=0.7+0.7i$. That modulated data is encapsulated in a vector as below: OR where $c$ is any constant number, let’s say, for example: $c=0.7$ ...
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1answer
31 views

Effectively extracting “real” data from a noisy dataset

This question was originally posed to Math SE. It was suggested that DSP SE would be more suitable. Background & Motivation: I have three lists of timestamps (UNIX timestamps, plus a subsecond ...
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0answers
18 views

Is it important to learn to find de period of the sum fo two DT signals?

The fundamental period (N) of the sum of two DT signals could be the LCM of the fundamental periods. However, this is not always the case. I would like to know wether, during engineering design or ...
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0answers
21 views

When is Markov a Martingale

I have two questions and I am very confused about the concepts Can a Markov process of order one also be a a Martingale? Is any Markov process of order one also a Martingale? For 1. I would say yes, ...
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2answers
242 views

Problems computing the DFT of finite length sequence

I am having trouble finding the same answer as the solution manual for this sequence. The problem asks to compute the DFT of $$ x[n] = \begin{cases} 1 & \text{for even } n \in \{0\...
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1answer
88 views

Linear correlation of two vectors?

We had a lecture on digital watermark detection using linear correlation. Here it was explained that the linear correlation of two vectors is equal to their scalar product divided by the dimension. So,...
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1answer
36 views

Relation between the matrix trace and the amplitude of each element

Assume a diagonal matrix $\mathbf X$ whose size $N\times N$ and its diagonal elements are $0.5 + 0.5i$, and the vector $\mathbf p$ of size $N\times 1$ whose elements have similar amplitude. I have ...
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2answers
256 views

DFT of pure sinusoidal wave

I'm writing a program in which you can synthesize waves by adding to a sound's Fourier transform, and then inverse the transform to get the modified sound. In order to do this, I need to know what to ...
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1answer
86 views

SNR for the magnitude of a complex signal

The SNR for a complex valued signal is presented by Dilip Sarwate in: https://dsp.stackexchange.com/a/42359 Following the same steps I would like to estimate the SNR for the magnitude of a complex ...
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1answer
29 views

I don't understand how the zeros are found of this 2nd order feed-forward filter

I realized someone asked a question about the exact same section of a book, which can be found here: Question about zeroes of simple 2nd order FIR filter. While the answer to this question looks ...
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2answers
177 views

Coordinate system in digital image processing

I am trying to understand image coordinate system where I read that it is reversed. So if $f(x,y)$ is actually $f(y,x)$ where $f(\cdot, \cdot)$ is the image. I am a bit confused and would appreciate ...
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1answer
60 views

Summations in Z-Transform

I'm currently working on a problem that involves a Z-Transform. Basically, the essence of the problem is that if: \begin{equation} H\left(z\right) = \sum_{n=0}^{N-1}h\left(n\right)z^{-n} \end{equation}...
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1answer
43 views

Adding attenuations in dB

I am writing a synthesizer. One part of the synthesizer specification says that different sources can specify an attenuation to be applied to a signal, and that these attenuations should be summed. ...
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0answers
101 views

Fourier Transform of an Exponential Sine Sweep

The Exponential Sine Sweep (ESS), according to Farina [1], can be described by the following formula: $$x(t)=\sin\left(\frac{2\pi f_1 T}{R}\left(e^{\frac{t R}{T}} -1\right) \right)$$ where, $t$ - ...
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0answers
73 views

Problem with natural logarithmic function in fixed point notation

I am facing a problem with natural logarithmic function in fixed-point notation. Let's say $$x = 0.54, \qquad\text{then}\qquad \ln(x) = -0.616186, \qquad\text{a negative number} \tag{1}$$ Then in ...
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1answer
85 views

FFT algorithm gives wrong signs on my output numbers

I was told from another stack exchange (stackoverflow and computer science stack exchange) to post here - so hopefully now i can get some help from people who understand the algorithm, but if this is ...
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1answer
108 views

Why Cramér spectral representation and not DTFT for stochastic process

In a lot of time-series analysis references I find (written by mathematicians or statisticians rather than engineers), I find the following signal decomposition for a stochastic process, termed the &...
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6answers
7k views

What is the first derivative of Dirac delta function?

Could you please help me in a simple way, what is the first derivative of a Dirac delta function? I found this answer: The informal answer is a positive Delta function immediately followed by a ...
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0answers
40 views

Setting some values in a vector to have specific values after multiplying with such matrix

Assume I have a known matrix $X$ of size, for example, $(16$ x $16)$, and a vector $z$ of size $(1$ x $16)$ where only some equispaced values are known, for example $1:4:16$ or values in locations $(...
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1answer
46 views

Solve equations with conjugates multiplications

If I have a two variables $x_1$, $x_2$ and two equations. In the first equation the first variable $x_1$ is multiplied with the conjugate of same number which is multiplied with $x_2$, and in the ...
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1answer
69 views

Finding the system output by convolution

pretty new to this concept, so do bear with me. A linear dynamic system is described by the following differential equation: Transfer function H(s) is calculated to be = I've already found the ...
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4answers
66 views

On the simplification using trigonometric functions

Assume I have a matrix $D$ whose its entries are as below : Where $A$ and $B$ can be written using using the trigonometric functions for (1) as: My question, Is it possible to simplify (1) more? ...
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1answer
118 views

Calculate autocorrelation of a sinus

I am a beginer in signal processing. I am currently studying a signal $s(t) = \cos(h(t))$, and I have to calculate its autocorellation. Since $h(t)$ is periodic of period $T_s$, I calculated the value ...
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2answers
101 views

Q format doubts

Is my understanding correct w.r.t Q format. signed Q15.1 format means, 15 bits for fractional and 1 for integer bits SO max range is - 2^15 / 2^15 to + (2 ^ 15 - 1) / 2^15 signed Q7.0 means, 7 bits ...
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2answers
35 views

Understanding Communication Theory Jargon for Statistical Linear Models

Coming from a statistical background, I understand the following formula: \begin{equation} y = X\beta + \epsilon \end{equation} as a linear regression model, with data input-output pair: $(X,y)$, ...
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3answers
2k views

Convolution of two rectangular pulses intuition

From what I understand the convolution of the impulse response of a system with the input to that system gives the output. Now if the impulse response is a rectangular function and the input is also ...
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1answer
142 views

Absolute value based AM envelope detection viewed in the frequency domain

In my understanding when a passband AM signal is demodulated with an envelope detector taking the absolute value restores the original baseband version of the signal and the low-pass filter removes ...
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1answer
155 views

Separation of signals without loss of information about the signals themselves

There is a signal that contains: Rectangular pulses on frequency of 1 kHz Rectangular pulses on frequency of 1.4 kHz Other noises. Required: Separate 1kHz pulses from everything else and measure ...
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1answer
70 views

Dummy DSP student request for IDTFT Calculation and Integration Process

We have one DSP course in biomedical engineering, so i have been for 10 years away from mathematical calculation and working in this years by programing by c and ..., so one of our exercise is to ...
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0answers
44 views

Solving equation with convolution

I have the measured signal $y(t)$ that can be modeled in the frequency domain as $Y(f)$: $$Y(f) = X(f)\cdot A(f) - [X(f)\cdot B(f)] \ast C(f)$$ where $\ast$ is the convolution. I know $A(f)$, $B(f)$,...
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0answers
563 views

Derivation of ZOH Discretization

I'm trying to understand the derivation of the zero order hold discretization method, and I have a couple of questions about some of the steps. I think I understand the first part, this is just the ...
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3answers
848 views

Resolution of Discrete Fourier Transform is 1/T - Mathematical proof?

In many articles I see that the frequency resolution of the Discrete Fourier Transform (DFT) equals Fs/N where Fs is the sampling rate and N is the total number of samples. Fs/N is equivalent to 1/T ...
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0answers
178 views

Power, energy and amplitude spectrum from fft in MATLAB - scaling

I've looked at previous answers and none of the answers seem to agree (perhaps I'm missing something obvious): I would like to know what the correct scaling to use is for amplitude, energy and power ...
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0answers
39 views

On $H_\infty$ norm for transfer function

For a given scenario in the context of control system, I'm trying to investigate how the $H_\infty$ norm can be calculated for a transfer function as follows: $$G(s)= \frac{w_n^2}{s^2 +2\zeta w_ns +...
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2answers
687 views

Differential Equations applications in Signal Processing [closed]

Does differential and partial differential equations subject has applications in Signal Processing? I have basic DSP background and know Matrix algebra and Fourier transforms very well, but I am not ...
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1answer
180 views

How to find the impulse response of filter that outputs the autocorrelation of the input?

I am trying to solve question 9.61 in the attached image. I guess, by evaluating the autocorrelation for a specific instance, say $ t = 0 $ , that the impulse response, $h(t)$, should be : $$ h (t)...
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2answers
128 views

Check whether a given equation is linear

$ a = (x, y) \in \mathbb{Z}^2 $ is given as a pixel. My equation in which $g$ is image(matrix) is defined as, $f(x, y) = 56g(x,y)+93g(x−1,y)+92g(x+1, y)−57g(x, y−1)+555g(x, y+1) $ How can we know ...
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1answer
48 views

how to set Equalizer's coefficient using generalized eigenvector.

In that paper https://jwcn-eurasipjournals.springeropen.com/articles/10.1186/1687-1499-2012-72 .. Equation 3, $J(w) = w^HBw/w^HCw$ .. $B$ and $C$ are matrices, $w$ the filter coefficients vector ...
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1answer
84 views

Intuitive understanding of frequency

I know that $f = \frac{\text{cycles}}{\text{unit of time}}$ and I know that period $T = \text{time it takes to complete one cycle}$. So if we take the definition $f = \frac{1}{T} = \frac{\text{1 ...
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2answers
52 views

Rearranging exp terms

I have some problems rearranging exp terms to receive a cos or sin. I guess the solution is silly but I‘m not able to spot my mistake. I know the solution (since i looked it up with wolfram alpha) ...
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1answer
29 views

Equation Derivation: Help with intermediate steps

As per the attached image above, I don't know how the equation in the first line transformed to the form in second line. I tried solving it by multiplication but the result was different.
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1answer
589 views

Laplacian of Gaussian Approximation and Gaussian Blur as the Solution of Heat Equation

While I was reading SIFT paper(Lowe, 2004), I came across the method that he apply "heat diffusion equation" to Gaussian function to derive that $$ \frac{∂G}{∂σ} = σ∇^2G $$ I searched Wikipedia and ...
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1answer
95 views

How to measure the distance between two circle-like-lines over 360 degree

I am looking for a slim and with that quickly programmable image processing strategy to measure the distance between two circles over 360°. It is important to notice that the circles are not perfect, ...
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1answer
180 views

How to prove Line Spectral Pair coefficient properties?

I am trying to understand the definition of LSP coefficients from http://www.dspcsp.com/pdf/lsp.pdf. We take the polynomial $a(x)$ of degree $M$ from the denominator of the LPC system function, and ...
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1answer
72 views

Explicit a succession using z inverse transform

Is it possible to explicit $y(n)$ of this mathematical succession in recursive form using z inverse transform?: $ y(0) = 1 \\ y(n+1) = 2y(n) + 3 $ I can't write ...
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0answers
39 views

Relationships between $f(t)$ signal and $f(t) \times \sin(\omega t)$ signal

Let's assume that there is a signal $f(t)$, $0\le t \le T$. Then, is there any special meaning of the signal $g(t) =f(t) \times \sin(\omega t)$? Like AM transmission. What is shown/heard when we play ...
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0answers
335 views

NMSE of complex numbers

Actually I am doing an RF measurement and this includes phase and power. i generate a list of powers ranging for example from (-50 to 40) dBm, while the phase rotate to make it for example spiral ...
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2answers
209 views

MLE parameter estimation — confusion regarding some terms in the pdf of complex normal r.v (Part 2)

This question is based on the application of the pdf which was an earlier question of mine asked here Confusion regarding pdf of circularly symmetric complex gaussian rv If $v \sim CN(0,2\sigma^2_v)$ ...