# Wasserfalldiagramm

This graph shows a waterfall plot over the region from 10Hz to the end of the measurement sweep. The plot uses logarithmically spaced data at 96 points per octave. To produce the waterfall plot click the Generate button in the bottom left corner of the graph area.

The labels at the sides of the plot show the time axis values

## How a Waterfall Plot is Generated

To understand what the waterfall plot shows and how its appearance is affected by the various waterfall controls it is helpful to first understand how it is generated.

Each slice of the waterfall plot shows the frequency content of a windowed part of the measurement's impulse response. 'Windowed' means we take the impulse response and multiply each sample in it by the value of a window, which is made up of a left side and a right side whose shapes we can choose (the window types are selected via the Spectral Decay entries in the 'Analysis Preferences'). Here is an example of an impulse response showing the original impulse, the window shape (in blue) and the windowed response.

Here is a zoomed in view of the early part, where the effect the windowing has on the windowed (lighter red) trace can be seen.

After the frequency content of the first windowed part of the impulse response has been obtained it is plotted as the first slice of the waterfall. The window is then moved along the response and the process is repeated for the next slice. The amount the window moves is determined by the time span of the waterfall and the number of slices that are to be plotted, so that the data for the last slice is from a section of the impulse response that is later than the first slice by the time range - for example, if the time range was 300 ms and there were 51 slices there would need to be 50 shifts of the window (the first slice has no shift) so each slice would be from data obtained after moving the window 6 ms along the impulse (300/50).

The window has a left hand side and a right hand side. In the plots above, the left hand window is a Hann type that ends at the peak of the impulse. The right hand side is a Tukey 0.25 (which means that for 75% of its width it is flat, then the remaining 25% is a Hann window). The overall width of the window (left side plus right side) determines the frequency resolution of each slice of the waterfall. The shape of the window, and particularly the shape and width of the left hand side, affects the way features of the response are smeared out in time.

To understand this, imagine a rectangular window and a perfect impulse, that has one sample at 100% and all other samples zero. As long as that single 100% sample is within the span of the window the frequency response will be a flat line. As soon as the left edge of the window goes past the 100% sample that slice and all slices after it will have no data in them (all the samples will be zero) so the waterfall would disappear off the bottom of the plot. Here is an example of such a waterfall plotted with a 100 ms left hand rectangular window.

That waterfall is, in the time domain, a faithful representation of how that perfect impulse response looks - and in general for any response a rectangular window gives the best time resolution, but that comes at a price. The price is in the frequency domain behaviour, i.e. the shape of the frequency response in slices of the waterfall. In real impulse responses, that are spread out over time, using a rectangular window creates a sharp step at the left hand edge of the windowed data. That sharp step causes ripples in the frequency response, obscuring the actual frequency content. The waterfall also has an initial period, equal to the width of the left hand window, where the slices are almost identical, creating a flat portion. Here is an example of a measurement with a 100 ms rectangular left hand window, despite its appearance it is the same measurement as shown at the top of this help page, only the shape of the left hand window has been changed.

To avoid the damaging effects of that sharp step in the windowed response, a tapered window is used to smoothly attenuate the samples, but now a feature that does actually have a rapid change in the impulse response will linger on in the waterfall, because it will not entirely disappear until the whole left hand window has gone past it. Here are the perfect impulse and the real measurement again, this time with a 100 ms Hann left hand window.

REW's waterfalls have been aimed at examining room resonances. To help make those resonances easy to see in the response, a wide left hand window is used - in REW V5.0 and earlier its width was half the setting entered as the Window time, and the right hand window had a width equal to the window time. However, that meant increasing the Window setting increased both the frequency resolution (the main reason for wanting a longer window) and also stretched the response out in time, due the increased left hand window width. That was not very helpful, as it meant the time range had to be increased to get back to a useful view of the behaviour.

After V5.0 the waterfall behaviour has been enhanced to improve control over its appearance and extend its use to include the analysis of drive unit and cabinet resonances. The left hand window width is specified independently, using a setting labelled Rise Time. Changing the Window setting only alters the Right Hand window, which means that the Window setting now controls only the frequency resolution of the waterfall - longer settings give higher resolution - without altering the waterfall's time domain behaviour. There are also controls to select how many slices the waterfall should have (up to 100) and to select the smoothing to apply to each slice.

In addition to the standard waterfall mode, which slides the window along the impulse response, there is a CSD (Cumulative Spectral Decay) mode, which anchors the right hand end of the window at a fixed point and only moves the left side, which is useful when examining cabinet or tweeter resonances over very short time spans. This does mean, however, that the frequency resolution reduces (and the lowest frequency that can be generated increases) as the slices progress, as each has a slightly shorter total window width than the previous slice.

## Waterfall Controls

The Slice slider selects which slice is at the front of the plot - as the slider value is reduced the plot moves forward one slice at a time. The trace value shows the SPL figure for the front-most slice, the corresponding time for that slice is shown at the top right of the graph.

The x, y and z sliders alter the perspective of the plot, moving it left/right, up/down and forwards/backwards respectively. The check boxes next to the sliders allow the perspective to be disabled in that axis. Disabling the x axis can make it easier to see the frequencies of peaks or dips. Disabling the z axis turns off all the perspective effects which makes the plot like a filled spectral decay. Here is the same plot as above but with the x-axis perspective effect turned off.

The waterfall allows another measurement's plot to be overlaid on the current measurement. The overlay is generated slice-by-slice, plotting a slice of the current measurement's waterfall, then a slice of the overlay, then the next slice of the current measurement and so on. N.B. before a measurement is available to overlay it is necessary to generate the waterfall data for it.

The overlay is selected using the Overlay selector. Measurements which do not have waterfall data are shown in grey in the selection list. To generate the data for a measurement select it as the current measurement and use the Generate button.

Transparency can be applied to the main plot, the overlay, or both. When transparency is set to 0% both plots are solid. In the image above the main plot is drawn at 75% transparency, allowing the overlay to show through. The transparency mode can be switched between main/overlay/both to ease comparison between the plots.

The Total Slices control determines how many slices are used to produce the waterfall. Fewer slices mean faster processing, but make it less easy to see how the response is varying over time.

The Time Range control determines how far the impulse response window is moved from its start position to generate the waterfall.

The width of the impulse response section that is used to generate the waterfall is set by the Window control (this control sets the Right Hand window width). The corresponding frequency resolution is shown to the right of the window setting. Longer window settings provide better frequency resolution.

The Rise Time control sets the time duration of the Left Hand window. Shorter settings give greater time resolution but make the frequency variation less easy to see. The default setting, 100 ms, is aimed at revealing room resonances. When examining drive unit or cabinet resonances with full range measurements a much shorter rise time would be used, 1.0 ms or lower, with time spans and window settings of around 10 ms. CSD mode is often more useful for such measurements as the later part of the impulse response can be noisy, obscuring the behaviour in the later slices. The 'rise time' terminology dates back to the late 80s and MLSSA. In MLSSA it referred to the 10% to 90% rise time of a left hand window formed by convolving a window function with a unit step (in essence, the step response of the chosen window function). The actual width of the window was much greater, depending on the window type - about twice the rise time for a Hann window, for example, or about 3 times the rise time for Blackmann-Harris. In REW the term is used to refer to the overall width of the left hand window, somewhat misusing it in the interest of retaining terminology that is in common use for CSD plots whilst adopting a definition that provides a clearer indication of what parts of the response lie inside and outside the chosen window settings. To obtain similar results to the MLSSA-style definition use an REW setting that is twice as long.

The Smoothing applied to the waterfall slices can be increased from 1/48th octave (the minimum, and recommended) to as high as 1/3rd octave.

Use CSD Mode should be selected if the later slices of the waterfall are contaminated by noise in the measurement. It would commonly be used when examining drive unit or cabinet resonances. CSD mode anchors the right hand end of the window at a fixed point and only moves the left side. This does mean, however, that the frequency resolution reduces (and the lowest frequency that can be generated increases) as the slices progress, as each has a slightly shorter total window width than the previous slice.

The control settings are remembered for the next time REW runs. The Apply Default Settings button restores the controls to their default values.