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74 views

Complex expression for periodic binary sequences

We have infinite binary sequences of type $$\langle g_n \rangle_{j=4}=\{0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,...\} \,;\, n\to\infty$$ where $j$ indicates the length of a period that starts/ends with a $1$; ...
2
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1answer
61 views

FFT for a single frequency

I was looking for a more efficient way of finding the magnitude and phase of a signal at a certain frequency without performing an FFT because it produces more information than I need and I came ...
2
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1answer
200 views

Check the orthogonality of two functions

This is a part of two Passband Signals. My question involves inner product. Is $$ \left < \cos(2\pi ft + \phi) , \sin(2\pi ft + \phi) \right> $$ where $\phi \in [-\pi, \pi)$ equal to zero, i.e. ...
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1answer
254 views

solve an difference equation's particular solution with unit step input

I have a question like this $y[n] - \frac 2 5y[n-1] = 2x[n]$ and $x[n] = 2u[n]$, the manual say that $y^p[n] = k u[n]$ and subtitute it into the equation will become $ k - \frac 25k = 4$. But I think ...
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1answer
141 views

How can I calculate the bode magnitude and frequency as well as their plots?

I've been trying to figure this problem out for a while now. I've been given a transfer function $$H(s) = \frac{s(s+100)}{(s+2)(s+20)}.$$ I'm supposed to calculate the bode magnitude and frequency for ...
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0answers
43 views

Discrete time fourier transform of partial sum

I came across the following property of the DTFT: $ \mathcal{F} \Bigg(\sum_{m=- \infty}^{n}x[m]\Bigg) = \frac{1}{1- e^{-j \omega}} X(e^{-j \omega}) + \pi X(e^{-j0}) \sum_{m= ...
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1answer
5k views

Prove of the Parseval's theorem for Discrete Fourier Transform (DFT)

If $x[k]$ and $X[r] $ are the pair of discrete time Fourier sequences, where $x[k]$ is the discrete time sequence and $X[r]$ is its corresponding DFT. Prove that the energy of the aperiodic sequence ...
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1answer
79 views

Find the non null complex Fourier coefficients

I'm doing my homework on signal processing in MatLab and I'm stuck on an exercise. I'm given this signal $x(t)=1 + 2\sin(12\pi t+\frac{\pi}{4})\cos(21\pi t);$ and I have to get the non null complex ...
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1answer
85 views

The science of pearson product moment correlation coefficient

I need to compare two sound signals for similarity, I took cross-correlation of both the signals and I got a cross-correlation signal, now I intend to use pearson correlation coeff formula to get the ...
2
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2answers
49 views

Express $\cos(5t)$ with the help of Euler formula

I can't figure out, how to express $\cos(5t)$ in the form $e^{j\omega t}$. I don't even know the right answer. How would you deal with this task?
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1answer
48 views

How does one verify if a vector is really recovered?

In compressed sensing, how to verify if a vector is really recovered or how does one plot the figures on recovery rate? Since in numerical experiments, there is always a difference between the ...
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1answer
118 views

How to determine the function of this “black-box”?

Imagine a black box which accepts a digital input signal which is a pure sine wave and outputs the area between this input signal and some constant data set over a sample window. Given the input ...
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1answer
76 views

response impulse function

my question is related to time series modeling in signal form,i have such question,suppose we have time series data $y_1........y_n$,how can we represent in impulse response form?as i ...
3
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1answer
153 views

Find the expression and the system impulse response

I've started to learn signal fundamentals and I have to do one exercise and I can't understand something. It is said that $$x[n]=1.5\cos(0.025 \Pi n)(u[n+40]-u[n-40]))$$ and that the signal $u[n-m]$ ...
2
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1answer
390 views

Relationship between DFT index values, frequency in a Fourier series and Hz.

I have a sound file recorded at 44.1 K samples per sec, and some FFT and IFFT algorithms. The sound file is a vector with about $ 2^{17} $ elements. My objective is to find which of the index values ...
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0answers
61 views

Pre-Emphasis of a signal

I'm trying to describe the process of Pre-Emphasis (of a signal) in my equations, but I don't know whether or not this makes sense. $Y[n] = X[n] - 0.95 \cdot X[n-1]$ Where Y = pre-emphasis after, X ...
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1answer
97 views

Discrete time signal and unit step

I've started to learn signal fundamentals and I have to do one exercise and I can't understand something. It is said that $$x[n]=3\cos(0.1 \Pi n)(u[n+55]-u[n-55]))$$ and that the signal $u[n-m]$ is a ...
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1answer
869 views

How to determine the step response using convolution of the signal's impulse response?

The step response can be determined by recalling that the response of an LTI to any input signal is found by computing the convolution of that signal with the impulse response of the system. ...
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1answer
44 views

Finding two highest peaks in a chunk of samples

Supose I have a list of 600 samples of numbers, and the histogram of the samples looks like the following: As can be seen, there are two 'mountains' of values around 16 and 48, and a outlier in 95. ...
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1answer
67 views

Cleaning a signal and computing period

I am working with a signal which is a periodic square signal with some kind of noise and some outliers. I would like to know which is the best solution in order to get the period and clean the ...
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2answers
130 views

What course sequence should I study in order to build a foundation for studying cryptography and signal processing?

I studied mathematics about two decades ago, but unfortunately, I remember little of it. I'm hoping to start studying cryptography and signal processing, but I'm not entirely sure what ...
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2answers
170 views

When writing out the wave equation isn't the phase shift just adding to the frequency?

I hope someone here could clarify for me. With the waveform function f()=(A)sin(wt + theta) where A=magnitude of the wave, sin= type of wave form, wt=frequency rads/sec, theta = phase shift ...
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1answer
223 views

Power spectrum for discrete signals.

If $x(t)$ is a real (aperiodic) power signal, i.e. \begin{equation} 0<\lim_{T\rightarrow\infty} \frac{1}{T}\int_{-T/2}^{T/2}|x(t)|^2 dt<\infty \end{equation} $x_T (t)$ is a truncated version of ...
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1answer
839 views

What does upside down “v” ($\wedge$) mean in this equation?

I have a simple question, but it is hard to google it. I have this equation here: $$y(t, x) = \sum_{i=1}^{d}(|x_i| \wedge t)^{2} $$ Here $x$ is a size $d$ signal and $t$ is just a scalar. I am not ...
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3answers
108 views

Recover filter coefficients from filtered noise

I have a digital signal which may be represented as noise filtered with an FIR (finite impulse response) filter. Let us suppose that the noise consists of pulses (nonzero samples on a zero ...
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1answer
83 views

Filter signal through convolution

I am a little bit unsure if I've set up the following problem correctly: Consider the signal $$f(t) = e^{-t}(\sin(5t) + \sin(3t) + \sin(t) + \sin(40t)) \quad 0 \leq t \leq \pi$$ Filter this signal ...
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1answer
80 views

(geometric) intuition of whitening

I found this http://cis.legacy.ics.tkk.fi/aapo/papers/IJCNN99_tutorialweb/node26.html But I still don't have an intuition of whitening. A diagonal covariance matrix means uncorrelated distributions, ...
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1answer
231 views

Does this curve tend to a square wave?

I have put some Mathematica code here: http://pastebin.com/cY6r7skS that uses this algorithm: $$y1 = Sin[x];$$ $$y2 = Sin[y1];$$ $$y3 = Sin[y1 + y2];$$ $$y4 = Sin[y1 + y2 + y3];$$ $$y5 = Sin[y1 + y2 ...
2
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2answers
361 views

Interchange of sums and integral

Suppose I have a function $x:\mathbb{R}\rightarrow\mathbb{R}$ such that is square-integrable: $$\int_{-\infty}^\infty|x(t)|^2dt<\infty$$ Suppose also that $x(t)$ contains no higher frequencies ...
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1answer
437 views

Can the differentiating and squaring process in the cochlea explain a reported dichotic stimulation experiment?

On this math.stackexchange on url What is Octave Equivalence? in an answer on the related ( octave equivalence ) question is stated: Mathematically, this signifies that the mammalian cochlea ...
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1answer
88 views

Region of convergence of Z-Transform connected area?

Shouldn't the Region of Convergence of the Z transform be a connected area ? In Oppenheim solution manual, I've found this answer of a question that asks to determine the different forms of the ...
2
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1answer
269 views

Upper and lower bounds of a ratio involving vector norms

I'm working on a signal processing problem and need to analyze the following expression $$ G = \frac{n}{\sum\limits_{i=1}^n |w_i|} \frac{ \sum\limits_{i=1}^n g_i w_i^2}{\sum\limits_{i=1}^n g_i |w_i|} ...
2
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2answers
327 views

Combining bins of FFT ouput

I was trying to combine output of a $2n$ point Real FFT to generate custom FFT bins. For example the FFT generates components at equally spaced frequencies $f_0,f_1,f_2 ...f_{n-1}$ $f_0$ = ...
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0answers
96 views

Explain the existence of limit in Persistence Excitation — mostly zero and non-existent?

Definitions Persistence Excitation on page 121 here or shortly here and here. A signal is PE if this limit exists $$r_u(\tau)=\lim_{N\rightarrow\infty}\frac 1 N \sum_{t=1}^{N} ...
2
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0answers
196 views

Lloyd-Max Quantizer Problem

Consider the function $z = x^2 + y^2$ , for $0 \leq x \leq 10$ and $0 \leq y \leq 10$. Take a regular discretization with steps $\Delta x = \Delta y = 0.1$ and its histogram as the probability ...
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0answers
179 views

What is the purpose and usage of convolution?

I am curious of what the purpose and usage of convolution are. Why is convolution created? In layman's term (and in mathematical term), what defines convolution?
5
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1answer
463 views

Zero-padding data for FFT

If I take a discrete Fourier transform of $\{ c_1, c_2, \ldots, c_n\}$ where $n$ is prime, I am rather limited in the FFT algorithms available to me and their performance. Additionally, having ...
2
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0answers
144 views

MFCC - Why 13 Coefficients

Basically I am trying to computer a MFCC and wondered if you can help. This is the FFT of 1 of the Frames (After I have multiplied the Hamming Window by the Mel Bank Filters) : Here is the DCT of ...
5
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2answers
201 views

Usage of inverse Laplace transform

At my current study level in college, use of inverse Laplace transform is not mentioned well - textbooks say "use tables." So, can anyone show me how to use inverse Lapalce transform? And also proof? ...
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1answer
3k views

Hamming Window - Understanding Formula

I'm trying to understand the Hamming window concept (Signal Processing) and the equation looks like the following: Where: N = number of Samples in each frame Y[n] = Output Signal X(n) = Input ...
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3answers
293 views

Removing noise when the signal is not smooth

Suppose we have (an interval of) a time series of measurements: We assume it can be explained as a "simple" underlying signal overlaid by noise. I'm interested in finding a good algorithm to ...
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3answers
189 views

Pre–emphasis - Signal Processing

I am trying to compute the Pre-emphasis of a signal and the formular is below: y[n] = x[n] - 0.95 x[n-1] Let: ...
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1answer
243 views

how can I get minimum error probability for this decision problem?

I have the decision problem for 4 hypotheses as follows: $$H_j: Y_k=N_k-s_{jk},\ k=1,2,\ldots,n;\ j=0,1,2,3.$$ where signals are $s_{jk}=E_0\sin(w_cT(k-1)+(j+\frac{1}{2})\frac{\pi}{2}).$ $$$$ In ...
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1answer
93 views

Number of Zero-crossing

I have written an algorithm that calculates the number of zero-crossings within a signal. By this, I mean the number of times a value changes from + to - and vise versa. The algorithm is explained ...
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2answers
85 views

Algorithm as Equation

I have developed an algorithm that counts the number of times a particular block (within a 2D Matrix) crosses zero. Here's an example: ...
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1answer
111 views

What does it mean to convolve a matrix with a kernel?

I have a Matrix, M, of dimensions width x height. The problem is to apply the [-1, 0, 1] filter along the x and y axis (i.e. convolve the image with [-1, 0, 1] kernel along horizontal and vertical ...
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1answer
269 views

How does convolution/deconvolution with gaussian affect signal mean

I have a discrete signal (an image actually), which I am convolving/deconvolving with a zero-mean Gaussian kernel. I would like some proof that these operations do not alter the signal mean. Well, it ...
0
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1answer
51 views

Nonlinear single-input single-output model

Assume a single-input single-output model $y = f(x)$ where time series $x = (x_0, .., x_{n})$ is the input, time series $y = (y_0, .., y_n)$ is the output and $f$ a function mapping $x$ to $y$. If ...
0
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1answer
114 views

ROC analysis: detection probability when false alarm is maximized.

I am wondering if detection probability always goes to 1 as false alarm probability goes to 1. Let's assume binary hypothesis problem: $\mathcal{H}_0: x(t) =n(t)$ $\mathcal{H}_1: x(t) = s(t) + ...
3
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1answer
98 views

A question about infinities and distribution functions

Let $\mathcal{P}_i$ be the set of probability density functions to which $f_i$ belongs, $(i=0,1)$. Furthermore assume that $$L(y)=\frac{f_1(y)}{f_0(y)}$$ is an increasing function for any chosen ...