# Questions tagged [proof]

The tag has no usage guidance.

86 questions
Filter by
Sorted by
Tagged with
1 vote
50 views

### Continuous Phase Modulation - Phase trajectory expansion

For continuous phase modulation (CPM), the circular phase trajectory is expressed as follows: $$\phi(t) = 2\pi h\int_0^t\sum_k \xi_k g(\tau - kT_s)d\tau + \Phi_0\tag{1.1}$$ Where: $h$ is the ...
• 3,406
53 views

### Proof of Hilbert transform of a real function $x(t)$ is generally a complex function

I am looking for a formal proof of the result: The Hilbert Transform of a real signal $x(t)$ is generally a complex signal. Can this be proven and if so how? Thank you.
57 views

1 vote
860 views

### Having difficulty checking for time invariance of discrete system

A system is given with the following equation: $$y(n) = 3y^2 (n-1) - nx(n) + 4x(n-1) - 2x(n+1)$$ I need to check for the linearity and time invariance of the system. By just looking at the equation I ...
• 211
2k views

### Proof regarding the periodicity of a continuous-time sinusoid after sampling

Question A continuous-time sinusoid $x_a(t)$ with fundamental period $T_p = \frac{1}{F_0}$ is sampled at a rate $F_s = \frac 1 T$ to produce a discrete-time sinusoid $x(n) = x_a(nT)$. Show that $x(n)$...
• 211
125 views

### Proof of weak stationary random process autocovariance always goes to zero?

Professor told me that if a random process is weak stationary, and it does not feature any periodic component, then its autocovariance always goes to zero. I can intuitively understand it, however, ...
• 21
265 views

### Proof that DFT does not require more than N points

I'm trying to show how the discrete Fourier transform (DFT) arises from the equation for the continuous-time Fourier Transform. I've run into an interesting caveat which I can't seem to find an ...
4k 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 ...
• 83
152 views

### Can $\delta(t+\infty)$ be a legitimate signal?

Mathematically speaking, when I try to use some signal to disprove a system is invertible, can I use the signal like $\delta(t+\infty)$ ($\delta$ representing the Dirac distribution)? For example, the ...
• 43
3k views

2k views

• 195
107 views

### Normalized LMS with a posteriori Error and Woodburry's Matrix Inversion

I was going through this paper and the author mentioned that we can prove the following using the Matrix Inversion Lemma (AKA Woodburry's Matrix Inversion Identity): Using matrix inversion lemma we ...
• 131
2k views

### DFT of time reversed signal

I was looking into proof and find something strange: The last part we obtain from DFT definition. $$X[k] = \sum^{N-1}_{n=0}x[n]W^{kn}_N, \quad\text{Where}\quad W^{kn}_N = e^{-j\frac{2\pi}{N}nk}$$ ...
• 57
1 vote
5k views

### Is $y[n] = n x[n]$ an LTI system?

How can I test if $y[n] = n x[n]$ is an LTI system? And any other system for that matter? For example, how come $y[n] = \left( \frac{1}{2} \right)^n u[n]$, where $u[n]$ is the unit step, is an LTI ...
• 25
726 views

### Mathematical relationship between highpass and lowpass filtering

Let $g, h_{HP}, h_{LP}: \mathbb{R} \rightarrow \mathbb{R}$ and $G, H_{HP}, H_{LP}$ denote their continuous Fourier transforms under the Fourier operator $\mathcal{F}$. Let $*$ denote the continuous ...
• 133
1 vote
54 views

### Is power density invariant to Fourier Transform? Does it hold through derivation?

Assumptions We have a finite discrete measurement. Introduction I had some troubles determining a power spectral densities needed for Wiener deconvolution. Looking through the continuous equations ...
• 111
993 views

• 13