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Why would the discrete fourier transform “see” signals like this? What is the origin of spectral leakage?

The discrete fourier transform of $x = (x_{0},\dots,x_{N-1})$ is defined as $\displaystyle X_{k} = \sum_{n=0}^{N-1} x_{n}\omega^{kn}_{N}$ where $\omega^{kn}_{N} = e^{-2\pi ik/N}$ and ...
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2answers
38 views

How to draw the Bode diagram for a given transfer function?

With this transfer function: $$G(s)=\displaystyle\frac{10(s+1)}{s(0.1s+1)}$$ I need to do operations to draw the Bode diagram manually I have this: $G(jw)=\displaystyle\frac{10jw+10}{-0.1w^2+jw}$ ...
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Definition of “uniformly regular” signals (as used in the book “Wavelet Tour of Signal Processing”)

The author uses the term "uniformly regular" and I get the idea of it's meaning through the context, yet the phrase is used as if could also have a precise mathematical meaning. Is there a definition ...
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9 views

How do I get around finding a digital filter which is a narrow bandpass with a small group delay?

I need to find a digital filter that meets the following criterion: Is a narrow bandpass (1/40 of normalized frequency width) As small as possible group delay (preferably less than 300 samples) ...
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1answer
25 views

Help for understanding Danielson-Lanczos lemma

The Danielson-Lanczos lemma is the basis for fast Fourier transform algorithms. Now, I do understand this step $\displaystyle X_{k} = \sum_{n=0}^{N-1} x_{n}\omega^{kn}_{N} = \sum_{n=0}^{(N/2)-1} ...
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0answers
21 views

Proper way to combine wavelet coefficients from multiple rounds of analysis

I am doing signal analysis for a time series and the assumption of signal is $$S = F + e$$ Where $S$ is the original signal, $F$ is the frequency component and $e$ is white noise (auto-regressive ...
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1answer
56 views

Looking for a nice expression of these functions in terms of trig functions

I have come across three sinusoidal functions f1, f2, and f3 which, up to scaling and translation, are very close to each other. When normalized and plotted together, they are hard to tell apart. ...
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18 views

Discrete-time lowpass filter with rapid response to significative changes in the input

I have a signal that looks like this: To obtain an average, I apply this formula y[i] = α * x[i] + (1-α) * y[i-1] With a relatively small value of α the ...
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0answers
16 views

Example 8.10.1 from Van Trees' Optimum_Array_Processing

According to Example 8.10.1, the task is to construct a beamspace matrix of standart linear array which consists of 10 elements. The beamspace matrix should have 3 column-vectors of Taylor series ...
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3answers
34 views

3 points on “horizontal” sinusoid, what is its period?

Imagine you have 3 points that are all a distance of 1 separated from each other. How do you find the sinusoid that goes through these 3 points if you also know that the sinusoid is not at a strange ...
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2answers
80 views

Physical interpretation of L1 Norm and L2 Norm

In signal analysis, students have no qualms about associating the L2 norm of a square integrable function f(t) as the energy associated with that signal. A good understanding of whether a function ...
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0answers
33 views

Subject to in equation

I have the formula below: $$\hat x_2=\arg\min\lVert x\rVert_2\quad\text{subject to}\quad A{x}=y.$$ But I didn't understand what was meant by "subject to" ? does $x$ is replaced by $x_2$? please can ...
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1answer
42 views

Causal non-causal sequences

This answer says that $Y = zX$ is a simplest example of non-causal system because it corresponds to $y_n = x_{n+1}$ and current output depends on future input. Yet, it is causal because both $x_n$ and ...
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1answer
41 views

Decomposition of $a\sin(\varphi t)+b\sin(\vartheta t)$ into AM and carrier

I feel like this should not be so hard, but I am somehow stuck. I would like to decompose the signal $$a\sin(\varphi t)+b\sin(\vartheta t)$$ into an amplitude modulation and a periodic carrier ...
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1answer
24 views

How to derive the process noise co-variance matrix Q in this Kalman Filter example?

How to understand the process co-variance matrix Q in the example below ( I extracted it from Wikipedia http://en.wikipedia.org/wiki/Kalman_filter ) "Consider a truck on perfectly frictionless, ...
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1answer
53 views

Parseval's theorem.

We consider two signal $h(t)$ and $g(t)$ such that $$\int_{-\infty}^\infty |g(t)|^2dt<+\infty$$ $$\int_{-\infty}^\infty |f(t)|^2dt<+\infty$$ Parseval's theorem states that: ...
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1answer
22 views

The bandwidth of the signal $x(t)$.

The bandwidth (B) of the signal $x(t)$ is the range of frequencies (measured on the positive semi-axis) in which $X(\omega)$ takes values ​​different from $0$. Very often $X(\omega)$ is different from ...
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0answers
42 views

Using l1 magic toolbox for compressive sensing : Positive definite matricies.

I'm trying to use l1 magic to reconstruct an image from a single pixel camera I've developed. The test functions used are random binary patterns projected onto the object scene, so each pattern is ...
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1answer
23 views

Is there a way to modify the exponential smoothing function to account for varying sample rates?

I am using a simple exponential smoothing formula to smooth a signal. X(n) = a * S(n) + ( 1 - a ) * X(n-1) However on certain setups, the sample rate is much ...
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1answer
22 views

Top and bottom power spectral density of a height profile

Imagine I have a simple 1D height profile which is NOT symmetric. Now, what is truly important for me is to know what are the frequency content of the top profile (i.e. a cut profile above the ...
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0answers
11 views

Is there a standard way for modeling a Kalman filter where the measurements are obtained from differences?

Consider for simplicity a Kalman filter applied to the one-dimensional state space model $x_{n}=f_{n}x_{n-1}+q_{n}$ $y_{n}=h_{n}x_{n}+r_{n}$ with white noise errors. Assume that $r_n=e_n-e_{n-1}$ ...
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1answer
20 views

find period of discrete cosine

let us consider following we should find period of this discrete signal,for periodicity we should have $x[n+kN]=x[n]$ or $10\cos(0.088\pi(n+kN) +\phi)=10\cos(0.088\pi n+\phi)$ or $0.088\pi ...
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0answers
19 views

Total change of a signal overtime

I have some signals whose analytic type I do not know. I can only sample them every 0.1 secs. I want to pick that signal that changes as little as possible. For example, between sin2t and sint I ...
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0answers
42 views

Covariance between real and imaginary parts of Fourier transform of a stationary time series

Since Fourier transform of a random stationary time series(in the case of existence) is not necessarily real, my question is what is the relation between the covariance of real and imaginary parts of ...
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0answers
27 views

Do 2 timeseries represent the input better than one?

I only have a very basic familiarity with signal processing and information theory so I'm sorry if this is a very straight forward question. I have a very brief input signal and two timeseries as ...
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1answer
28 views

Discrete Time Fourier Transform of a real signal

I want to prove that if we have a real signal x[n] then for the DTFT it is applied that we have an even symmetry: | X(Ω+1/10) | = | X(-(Ω+1/10)) | (I mean the ...
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1answer
45 views

Image Reconstruction:Phase vs. Magnitude

Figure 1.(c) shows the Test image reconstructed from MAGNITUDE spectrum only. We can say that the intensity values of LOW frequency pixels are comparatively more than HIGH frequency pixels. $$ ...
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0answers
37 views

Expectation and convolution question.

I am learning in an image processing course, and the professor did the following: As part of a derivation, has this: What I do not understand, is how he was able to remove $r(i,j)$ to the ...
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1answer
32 views

Blind deconvolution of a function convolved with itself

I have a function/vector $f$ that I know is the result of an unknown function $g$ convolved with itself: $f = g \ast g$ Is there any way to do a blind deconvolution on $f$ with this constraint?
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1answer
33 views

A relationship among multiple periodic arrays

There are N periodic arrays ai[n] with period Ti, respectively, where i=1, 2, … , N. Each array has a property that a[n]=1 when n=k*T where k is integer, otherwise a[n]=0. Then a new array is created ...
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1answer
40 views

Vctor of sine signal [closed]

How to create a vector of sine signal (I am looking for an example such a function which can create such vector) thanks
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0answers
30 views

On the truncation of spherical harmonics

Suppose there is a function $f(\theta,\phi)$ defined on the surface of a sphere, and $\theta$ and $\phi$ are the polar and azimuthal angles respectively. Similar with the fact that a function defined ...
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1answer
34 views

Signal processing and properties question

We have the relationship between input and output: $Y(k)=|x(k+1)| + x(k)+ kx(k-5)$. Find the output of the system when $x(k)=d(k)$. What does this output represent? Show if the system is linear, ...
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0answers
24 views

Solution to iterative equation with floor operation

This question is motivated by the following signal processing problem. Suppose there is a source, which produces vectors of data of length $N_s$, and a filter (or other subsystem) that accepts ...
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1answer
35 views

How can I plot this signal

What will be the value of this signal (specified as impulse train) , say for values of t from 0 to 6 $$g(t)=\sum_{k=- \infty }^\infty \delta(t-2k)$$
3
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1answer
115 views

Approximate Periodic Function by shifting Basis Functions

Given a periodic "Target Function" $F(t)$ a set of $N$ periodic "Basis Functions" $B_i(t)$ of arbitrary shape All functions are defined on the same interval $T$. I am allowed to shift ...
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0answers
41 views

Function with bounded derivative as ODE

Given a function $x(t)$, I am looking for a function $y(t)$ which closely follows $x(t)$ except that its derivative must be bounded by a constant $c$, i.e. $\dot{y} \leq c$. Is there a way to describe ...
2
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1answer
62 views

Moving average as ODE

Is it possible to represent or approximate the moving average $m(t) = \frac{1}{w}\int_{t-w}^t x(\tau) d\tau$ of a function $x(t)$ as a set of ordinary differential equations $\dot{y} = \ldots$? I am ...
2
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1answer
46 views

Trigonometric regression

What methods are performed for regression with trigonometric functions? E.g. : Sequence: $$-1, 0, 1, -1, 0, 1, \text{.....}$$ Regression: ...
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1answer
40 views

Fourier and $Z$ transform of a signal?

We have $$X(k)=4[u(k-2)-u(k)* d(k-3)]$$ I need to find the Fourier transform,$Z$ transform,as well as dhe magnitude and phase spectra. First of all I think that I need to convert the $u(k)$ and ...
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0answers
18 views

How to evaluate the difference between two classes of data which are highly overlapped

I’m trying to implement a classifier based on a dataset comprising two classes of high dimensional time-series observations (the values of the two classes of observations are highly similar). I ...
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1answer
40 views

How do we determine the duration of a fundamental frequency using the DFT (or FFT)?

I'm still in the process of learning the details of the DFT (and FFT) and I've just made a test .wav file in Audacity by joining 3 one-second sine waves together. .wav file 1 = 440 Hz, sample rate ...
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0answers
34 views

Plotting the frequency spectrum of a signal

I've found this algorithm here on Mathematica.SE to plot the frequencies of a signal using Fourier. It works beautifully, but I'm having some trouble understanding ...
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1answer
40 views

Impulse response and z transform question?

We have $g(k)=\{ [(1/5)^k]u(k)\text{ for $1 \le k\le3$ and $0$ for other }k\}$ The input is $x(k)=\delta(k) +3\delta(k-1)+ \delta(k-2) $ Using Z transform we have to find the output $y(k)$ and the ...
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1answer
12 views

Representing a real sampled signal with N samples as a complex sampled signal with N/2 samples

I am studying the discrete Fourier transform, and in its most basic definition it is an invertible linear transformation on the complex numbers. From Wikipedia: The sequence of $N$ complex numbers ...
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1answer
26 views

Differential equations signal theory question?

We have the system in the figure.I have to find the differential equation that describes the system.Now,the equation is x(k-1)-y(k-1)=y(k),but I don't understand why.I mean I have studied systems and ...
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1answer
49 views

Essential mathematics for Image Processing

What are the most essential mathematical concepts one has to be familiar with for succeeding in the field of Image/Signal Processing and Machine Learning. I am somewhat familiar with Tensors, ...
2
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1answer
24 views

Problem about average of cos square (nt) where n is arbitrary

I often see people just say time average of cos^2(nwt) is 1/2, I want to know in what cases this is not valid? w is just the frequency, can be assumed as a constant. Assuming you are always ...
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0answers
30 views

Expected Value With Signum

So I am trying to show convergence of a filter, and in order for it to converge, I need the following condition to hold: $ E\{ \mathbf{s} x sgn(\mathbf{h}^{T} \mathbf{s} x) \} \; \alpha \; E \{ ...
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1answer
59 views

If $tf(t)=tg(t)$, where $f(t), g(t)$ are distributions, then $f(t)=g(t)+\lambda \delta (t)$

Using Dirac distribution properties, prove that if $tf(t)=tg(t)$, where $f(t), g(t)$ are distributions, then $f(t)=g(t)+\lambda \delta (t)$ for some $\lambda\in \mathbb R$. If someone knows please ...