Skip to main content
9 votes

Factor $|a|^{-1/2}$ in definition of mother wavelets

My answer is for real scale $a$ and the fact that wavelet transform is usually defined in $L_2$ with norm $$||\Psi(\tau)|| = \int_\mathbb{R} \Psi(\tau)\Psi^*(\tau)\mathrm{d}\tau $$ So $$||\Psi_{a,t}...
AlexTP's user avatar
  • 6,605
9 votes
Accepted

Fixed point scaling; float -> Q31 -> float

Fixed point processing is very difficult (unless it's a very simple algorithm and well behaved data). Here is roughly how this can be done. For simplicity I will assume that the fixed point data type ...
Hilmar's user avatar
  • 45.6k
7 votes

Fixed point scaling; float -> Q31 -> float

OK -- I see your problem. You're multiplying $a$ and $b$, by first bringing them into Q31 then bringing them out. $a$ and $b$ happen to be 6 and 10, but let's not worry about that at the moment. ...
TimWescott's user avatar
  • 12.9k
6 votes
Accepted

Theoretical Maximum of DFT

A simple bound is just the number of samples. Indeed, for $$ X_k = \sum_{n=0}^{N-1} x[n]e^{-2i\pi kn/N}$$ then obviously: $$ |X_k| \le \sum_{n=0}^{N-1} \left|x[n]\right|\left|e^{-2i\pi kn/N}\right|$$...
Laurent Duval's user avatar
4 votes
Accepted

Overlap/add time-domain audio frames: How does normalization/scaling work with overlap greater than 50%?

Let's assume continuous time (rather than discrete time). If you do not process the windowed data at all, you would like the output (the sum of the windowed frames) to be equal to the original signal....
Olli Niemitalo's user avatar
4 votes

why should I scale the fft using 1/N?

FFT is a fast way to compute DFT. Hence the scale factor $1/N$ belongs to the DFT (specifically the inverse DFT in MATLAB ifft() function). As Marcus has already pointed out; it's arbitrary to put ...
Fat32's user avatar
  • 28.3k
4 votes
Accepted

Software-Emulated Amplifier: Cracking while change Volume

I’ve heard it called popcorn noise before. Think of the output signal as the product of the input signal and the volume signal. A discontinuity in the volume signal causes a discontinuity in the ...
Dan Szabo's user avatar
  • 1,038
4 votes
Accepted

Understanding the multiply by 2 factor in scaling the DFT magnitude

Scaling is indeed done for the first two reasons, and the scaling chosen is based on what units we want the output of the DFT to represent. The analysis for the OP's item (3) is incomplete, leading to ...
Dan Boschen's user avatar
  • 52.3k
3 votes

why should I scale the fft using 1/N?

Your $dt$ has an implicit $1/N$ in it: $$ dt \frac{time}{sample} = \frac{ T_{DFT} }{ N } \cdot \frac{ \frac{time}{frame} }{ \frac{samples}{frame} } $$ That's why it works. My preference is strongly ...
Cedron Dawg's user avatar
  • 7,590
3 votes
Accepted

why should I scale the fft using 1/N?

OK, let us go for a 2-point DFT. It should be noted that, depending on the software used, scaling can be different, and ought to be checked. The standard unscaled version does multiply the input ...
Laurent Duval's user avatar
3 votes
Accepted

How do scaling of the coefficients for the poles of a biquad filter affect the gain?

Sorry, that doesn't work. You have to scale all numerator (the "b") or all denominator coefficients (the "a") at the same time. That doesn't work for the denominator since you all practical ...
Hilmar's user avatar
  • 45.6k
3 votes

Fixed point scaling; float -> Q31 -> float

Here is a higher level summary of Q notation that will help demystify the other good answers: The post mentions "Q31" as a signed binary number, in this case with 32 total bits, and scaled ...
Dan Boschen's user avatar
  • 52.3k
3 votes
Accepted

Differences in PSD for windowed vs non-windowed spectra

The equivalent noise bandwidth (ENBW) for a given window $w[n]$ is the reciprocal of the window's processing gain and given as: $$ENBW_{bins} = N\frac{\sum(w[n])^2}{(\sum w[n])^2} \tag{1} \label{1}$$ ...
Dan Boschen's user avatar
  • 52.3k
2 votes
Accepted

Implementation of IIR Filter in dsPIC 33EP - Scaling

The tf2sos function takes an input filter of order $N$, given by $H(z)=\frac{\sum_i^{N} b_iz^{-i}}{\sum_i^{N} a_iz^{-i}}$ and returns coefficients for $N/2$ second-...
Maximilian Matthé's user avatar
2 votes

Theoretical Maximum of DFT

Parseval's theorem states that the energy in the output of the DFT is proportional to the input energy. That proportionality factor depends on how you define the DFT (whether you divide by the length, ...
Marcus Müller's user avatar
2 votes

Theoretical Maximum of DFT

An upper bound for the absolute value of the DFT coefficients can be derived as follows: $$|X[k]|=\left|\sum_{n=0}^{N-1}x[n]e^{-j2\pi nk/N}\right|\le \sum_{n=0}^{N-1}|x[n]|\le N$$ if we assume that $...
Matt L.'s user avatar
  • 90.5k
2 votes
Accepted

How to "scale" the FFT when using it to calculate discrete convolution?

this formula $FFT^{-1} \left( FFT \left( F \right) \odot FFT \left( G \right) \right) \stackrel{!}{=} F \ast G$ is not always true, it's true when you scale at the inverse operation as you mentioned ...
Det's user avatar
  • 46
2 votes
Accepted

How to scale a signal to get desired variance

In a practical setting to adjust the variance (thereof the power) of a random process, you could use the following to get what you want. Let the variance of a given RV $X$ be $$\text{Var}\{X\} = \...
Fat32's user avatar
  • 28.3k
2 votes

What's the minimum decibel value?

By any reasonable metric, the dB value for zero amplitude is minus infinity.
Dan Szabo's user avatar
  • 1,038
2 votes

Factor $|a|^{-1/2}$ in definition of mother wavelets

Wavelets play differents role in functional spaces, especially as unconditional bases (see What are unconditional bases and which wavelets have this property?). In $L_p$ spaces, if $|\psi|^p$ is ...
Laurent Duval's user avatar
2 votes
Accepted

Time scaling and shifting of delta function

Note that the time-scaled and shifted Dirac delta impulse $\delta(2t-1)$ is non-zero where $2t-1=0$ is satisfied, i.e., at $t=\frac12$. It's area is indeed $\frac12$ because we have $$\delta(at)=\...
Matt L.'s user avatar
  • 90.5k
2 votes

How to improve accuracy while converting floating point coefficients to fixed point in the case of an all pole IIR filter

Where are the poles of your filter? If they are close to the unit circle, then quantizing them can cause some of them to cross beyond the unit circle, causing the quantized filter to be unstable. You ...
Ben's user avatar
  • 3,777
2 votes

Impulse response of a time scaling system

The input-output relation of such a system can be written as $$y(t)=\int_{-\infty}^{\infty}\delta(t/2-\tau)x(\tau)d\tau\tag{1}$$ Note that the system is linear but time-varying, and such systems can ...
Matt L.'s user avatar
  • 90.5k
2 votes
Accepted

What's the minimum decibel value?

Remember that decibels are a ratio, fundamentally, on a log scale. As a ratio, there are no inherent bounds, and there is no inherent reference. When you say that 0 dB is the lowest amount we can hear,...
Nigel Redmon's user avatar
2 votes

How to evaluate fixed-point implementation of LMS filter is correct?

I'm not an expert on the LMS algorithm. Perhaps you should add a link to the algorithm description so we can help you. However, I have adapted a lot of algorithms to fixed-point implementation, so I ...
Ben's user avatar
  • 3,777
2 votes
Accepted

Algorithm that enlarges the image to a resolution of $2N \times 2N$ using DFT operations

I am copying my answer from Applying 2D Sinc Interpolation in the Fourier Domain (DFT / FFT). Given a Matrix $ A \in \mathbb{R}^{m \times n} $ in order to interpolate it into a grid of size $ k \times ...
Royi's user avatar
  • 19.8k
2 votes

Differences in PSD for windowed vs non-windowed spectra

See this answer I gave recently that discusses this topic. The scaling of the PSD is not a once and for all scaling, as how you estimate the PSD determines the appropriate scaling. The scaling of the ...
Baddioes's user avatar
  • 1,135
1 vote

Time scaling and shifting of delta function

Without needing to remember formulas about Dirac Deltas (a.k.a. impulses) as in Matt L.'s answer, here is how to proceed. For ordinary everyday use, impulses are defined by what they do in integrals, ...
Dilip Sarwate's user avatar
1 vote

Time scaling and shifting of delta function

For a classical function, you can indeed shift and stretch $\sigma$ for their representation. But as long as you are doing a comparison with another function, under an integral sign, like convolution ...
Laurent Duval's user avatar
1 vote

Pixel Binning: Effect on SNR for Hardware vs. Software Binning

If you have access to the raw pixel data, you can implement binning in software too. They should be identical upto the step-size of the ADC, by which the quantized analog sum of pixels may differ from ...
Fat32's user avatar
  • 28.3k

Only top scored, non community-wiki answers of a minimum length are eligible