Signal Processing Asked by mickkk on December 25, 2020
Given an $L$ order cosine window, it is possible to show that the width of the main lobe is given by:
$$omega_w = frac{2 pi L}{(2N+1)}$$
Where $L$ is the order of the window, $N$ is the maximum harmonic in the signal and therefore $2N+1$ is the number of samples taken over a period. If we consider the relationship
$$Omega = w T_s$$
then the width of the window in the frequency domain is given by
$$Omega_w = frac{2 pi L}{(2N+1)T_s}=frac{2 pi L}{T_0}$$
Where $Ts$ is the sampling time and $T_0$ is the observation interval.
The last formula clearly shows that it is possible to reduce the width of the main lobe either by oversampling by an oversampling ratio of $L$ or by increasing the observation interval by a factor of $L$.
Let’s assume the following:
Now, if we call $P(jOmega)$ the spectrum of the sampled signal, $P(jOmega) = W(jOmega) star S(jOmega)$ where $W$ and $S$ are the Fourier transforms of the window and of the signal, respectively and $star$ is the convolution operator.
What would $P(jOmega)$ look like? In the book I use, it looks like this:
STATEMENT A:
In my understanding at least, only $c$ is real (only $c$ is a replica of the real spectrum of the signal) while $a$ and $b$ are a byproduct of the convolution with the window due to its main lobe’s width. In general, the book says, it is possible to say that if we are using an $L$ order cosine window, $2L-2$ phantom (or fictitious if you will) components will appear.
STATEMENT B:
If we increase $T_0$ by $L$ then the width of the Hanning window becomes the same as the one of the rectangular window ($Omega_w = frac{2 pi}{(2N+1)T_s}$), therefore the fictitious harmonic components should not appear. But isn’t this the same as increasing the number of samples by a ratio $L$?
I would like to understand if the two above statements are correct, in particular if the origin of the fictitious harmonic components is indeed the convolution and how one can avoid these components.
The book that I mention is Digital Signal Processing for Measurement Systems by D’Antona and Ferrero.
The magnitude and number of the lobes/harmonics is beyond my knowledge but you always obtain them when you window a signal with a finite window. No matter what windows you choose.
With some windows like the cosine family ones you have an efficient way to compute the convolution in the frequency domain, but if you compute it in the time domain the result is the same.
Hope it helps a bit
Answered by OldApprentice on December 25, 2020
1 Asked on November 25, 2021 by nagabhushan-s-n
4 Asked on November 23, 2021
1 Asked on November 23, 2021 by alon-shmiel
0 Asked on November 23, 2021
discrete signals extrapolation feature extraction matlab sampling
1 Asked on November 23, 2021 by image-check
image processing noise pixel wise operation signal analysis snr
1 Asked on November 23, 2021
0 Asked on November 23, 2021
1 Asked on November 21, 2021
filter design filters finite impulse response window functions
5 Asked on November 21, 2021 by g6kxjv1ozn
2 Asked on November 21, 2021
2 Asked on November 21, 2021 by mczhang
bayesian estimation estimation gaussian kalman filters linear systems
1 Asked on November 21, 2021
discrete signals filter design filters fixed point lowpass filter
2 Asked on November 15, 2021 by cedron-dawg
1 Asked on November 11, 2021 by ali-khalil
1 Asked on November 11, 2021
1 Asked on November 8, 2021 by rotano
discrete signals signal analysis source separation stationary
Get help from others!
Recent Answers
Recent Questions
© 2023 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP