6

[EDIT] In 1991, Nasir Ahmed wrote: "How I Came Up with the Discrete Cosine Transform". Interesting to read, on how he was inspired by Chebyshev polynomials, and on how he didn't get funding, for a tool at the heart of JPEG and MP3. Natural images are not very stationary, but locally, their covariance is often modeled by a first- or second-order ...


3

Dirac delta function has a continuous argument, but Kronecker delta function has a discrete argument. Your example is a discrete signal so Kronecker delta is used.


3

I believe that h(-t) means a "time-reversed" version of h(t). Your command: 'y = conv(r,-h);' computes the convolution of 'r' and negative 'h', and you don't want that. I think you want: y = conv(r,conj(fliplr(h)));


3

As it was already posted multiple times: The problem comes from an inaccurate definition of correlation in your application. The Pearson correlation coefficient does require the data to be centered, ie the mean must be subtracted normalized, ie the data must be divided by the standard deviation This centering and normalization must be done for the mask ...


3

Convolution: $$ y(t) = h(t) \circledast x(t) = \int\limits_{-\infty}^{\infty} h(u) \, x(t-u) \ \mathrm{d}u $$ Cross Correlation: $$ R_{xy}(t) = \int\limits_{-\infty}^{\infty} y(u) \, x(t+u) \ \mathrm{d}u $$ The difference between the two is effectively the sign on $u$ in $x(t-u)$ in the integral. That correlation is like convolution but with one of the ...


2

There are two issues / concerns with this approach in that you may not be getting what you want. The primary one is the equivalent of post detection averaging by using the post-processed results from a spectrum analyzer, and the second is that the result of your complex conjugate multiplied FFT is actually in the time domain since you started in the ...


2

That is because correlation (and convolution) are not meant to "match" exactly a given pattern. They are multiplicative operators in their nature so they are strongly related to signal amplitude if you multiply the reference signal by N, the output gets twice bigger. for instance your operator will return a peak twice higher when encountering this piece of ...


2

You picked a tough example. Short answer: Change your "0"s to another value, e.g., 2, and it should work much better. What's really happening: your signals are not zero mean, correlation requires to center the signals (i.e., subtract means). Example, since it's easer to understand in 1-D: say you want to find the pattern p=[0,2,2,0] in the sequence s=[2,...


2

This might be hard with trying to illustrate this way but here I go. Take your original image and put the filter in the upper left corner like so: +-------+-------+---+---+ | 1 [0] | 1 [0] | 1 | 1 | +-------+-------+---+---+ | 1 [1] | 1 [1] | 1 | 1 | +-------+-------+---+---+ | 0 | 0 | 1 | 1 | +-------+-------+---+---+ | 1 | 1 | 1 | 1 | +----...


1

In other words, are ACF plots sufficient for determining whether a signal is a white noise? A plot is never sufficient. An impulse-shaped ACF is. (A plot can only ever be an estimate based on observation; to find the true ACF, you'd have to observe the process for infinity, which you can't.) You can only observe that the plot approximates the ideal ACF ...


1

The difference here is that statistical autocorrelation assumes a stationary power signal (so basically an infinite periodic signal) and does a normalisation to $[-1,1]$ and that autocorrelation just takes the finite signal as it is and does the classical convolution of the signal with itself, leading to just one point of perfect overlap and decreasing ...


1

Even if your amplified signal matched the quantization noise floor of the ADC (representing a 3 dB degradation) you would still achieve a significantly larger and clearer peak from what you are showing. The search bin size for 1 C/A code duration of 500 Hz is a good choice for the largest step (since the correlation versus frequency offset would follow a ...


1

If the constellation at the tarsmitter is symmetric about 0 on the real line then you can simply threshold the output between -1 and +1 with 0 as the threshold. So anything above zero is a logical 1 and everything below zero is a logical 0. This detection is rule is optimal when the noise is additive.


1

CRC is a "definite" indicator of the channel impacting the bits and error in received symbol. So deploying a CRC check would definitely help. Length of CRC would depend on the overall system and data packets.


1

The definition of auto-correlation depends on the type of signal. For random processes, the auto-correlation function is defined by the expectation given in Eq. $(1)$ of your question. For deterministic signals, there are two definitions, depending on whether the signal is an energy signal (i.e., has finite energy), or a power signal (i.e., has finite power ...


1

Energy signals (such as an exponentially decaying pulse) are bounded in total energy as time goes to infinity in which case you can use the first definition. Power signals (such as a sine wave) have finite energy over finite time intervals but infinite energy for all time (power is energy/time) so the first equation will not converge in that case.


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