My english is not so good. the time that the waterfall show is 2,94 milliseconds and the waterfall contain 100 segments. each segment(slice) have a line. each line is time 2.94 ms /100 0.0294 ms .with the rise time 0.5 ms there are 4 slices see at near same level at 20 khz. and 4 slices are time 0.0294*4= 0.1176 ms. but 20 khz have a period time of 0.05 ms. so i think when the decay time is so slow as the waterfall show this speaker can not play 20 khz. with rise time of 0.2 or 0.1 sec it look more as reality because when look at the first 4 slices after the impulse the decay time is much more
I suspect some of us may have some confusion or misunderstanding as to how waterfall plots are generated. I would like to describe, as I understood it, how REW generates its waterfall plots and some subtleties of the process. Hopefully, this will help those readers of this thread less familiar with FFT based signal analysis. Apology to
@bennybbbx if I am mistaken.
This is from the REW help file. The red curve is the impulse response from a REW measurement, and the blue curve is the "window" used to compute each slice of the waterfall plot. Look closely and you'll see 3 vertical dash lines (I'll call them left, middle, right). The "window" length is the length of time between the left vertical line and the right vertical line. The "rise time" is the length between the left and middle lines.
The window length determines the frequency resolution of each slice (Δf = 1/T, where T is the window length and Δf is the spacing of the frequency bins and therefore, frequency resolution). We cannot determine frequencies at an instantaneous time. There is always a trade-off between time resolution and frequency resolution. We need to know how the signal is changing to measure frequency, and we need multiple samples of the signal spread out in time to do that. The more samples we have, i.e. the longer the sampling duration, the better the frequency resolution but the poorer the time resolution.
Each of the slices in the waterfall plot is the frequency spectrum of the impulse response multiplied with the window. If the window is zero at a certain time location, the signal at that time location is excluded because the product of the window and signal is zero. Therefore, when calculating the frequency spectrum the slice, only the portion of the signal is considered when the window is non-zero.
The slices are generated by moving (sliding) the window to the right in time steps. The time step is the time range of the plot divided by the number of slices (i.e. a time range of 10 ms and 50 slices will give a time step of 0.2 ms). Notice that time step is
NOT window length. The duration of the signal used to compute each slice is usually a lot longer than the time step.
The window consists of three regions. There is an initial gradual rise in the shape of the rising half of a raised cosine (half Hann window), a constant horizontal flat line of 1, and a gradual drop at the end in the shape of a falling cosine. The rise time is the length of time of the initial rise from 0 to 1. Therefore, a shorter rise time will give a sharper response to a spike as it moves out of the window quickly as the window slides across the signal.
So why do we need the gradual rise and drop in the window instead of just sharp edges (i.e. a rectangular window)? This is because we want to minimize "spectral leakage". Below is an example using a rectangular window (shaded green). Only the signal inside the green box is used to calculate the frequency spectrum (figure on the right).
However, when we are performing a finite length discrete Fourier transform (as in all FFT), the math actually treats the signal as continuously repeating with a period of the total sampling duration (window time), see below.
If the signal at the start of the window is not exactly the same as at the end, there will be a discontinuity. This type of discontinuity is "spectrally rich", and add spectral contents not present in the original signal. That's why we use smoothly rising and falling windows to smooth out these discontinuities at the start and end of the signal. Therefore, we see a lot of jaggedness in the plots with short "rise times".
Ultimately, the purpose is to evaluate the decay time of the amplitude envelop of an oscillation at the frequency of interest. This decay time number is not very meaningful, and difficult to define, if we are looking at time scales of only a few periods of the oscillation.