Is there a technical term for this simple method of smoothing out a signal?

Question

Firstly, I am new to DSP and have no real education in it, but I am developing an audio visualization program and I am representing an FFT array as vertical bars as in a typical frequency spectrum visualization.

The problem I had was that the audio signal values changed too rapidly to produce a pleasing visual output if I just mapped the FFT values directly:

So I apply a simple function to the values in order to "smooth out" the result:

// pseudo-code
delta = fftValue - smoothedFftValue;
smoothedFftValue += delta * 0.2; 
// 0.2 is arbitrary - the lower the number, the more "smoothing"

In other words, I am taking the current value and comparing it to the last, and then adding a fraction of that delta to the last value. The result looks like this:

So my question is:

1. Is this a well-established pattern or function for which a term already exsits? Is so, what is the term? I use "smoothing" above but I am aware that this means something very specific in DSP and may not be correct. Other than that it seemed maybe related to a volume envelope, but also not quite the same thing.

2. Are there better approaches or further study on solutions to this which I should look at?

Thanks for your time and apologies if this is a stupid question (reading other discussions here, I am aware that my knowledge is much lower than the average it seems).

Answer 1

What you've implemented is a single-pole lowpass filter, sometimes called a leaky integrator. Your signal has the difference equation:

$$ y[n] = 0.8 y[n-1] + 0.2 x[n] $$

where $x[n]$ is the input (the unsmoothed bin value) and $y[n]$ is the smoothed bin value. This is a common way of implementing a simple, low-complexity lowpass filter. I've written about them several times before in previous answers; see [1] [2] [3].

Answer 2

Warning: include some history, old papers (I love them) and punch cards!

You used, with $a=0.2$ the form: $$y(n) = y(n–1) + a[x(n) – y(n–1)]\,,$$ sometimes written as: $$y(n) = ax(n) + (1 – a)y(n–1)\,.$$

The first above version is less natural, but it avoids one multiply, and is somehow more efficient. Both formulae yield a linear, causal and infinite impulse response filter. Story goes back to and through Poisson, Kolmogorov-Zurbenko Adaptive Filters, Brown (Statistical Forecasting for Inventory Control. McGraw-Hill, 1959), Holt (1957) and Winters (1960). It is implemented as a recursive filtering scheme known under different names across the literature:

• first order exponential averaging low-pass filter,
• exponential averager,
• exponential smoothing,
• exponential moving average (EMA),
• exponentially weighted moving average (EWMA),
• Brown's Simple (linear) Exponential Smoothing (sometimes called SES),
• ARIMA(0,1,1) model.

The "exponential" in the name is related to the impulse response with the geometric progression, that samples an exponential decay: $h[n]=(1-a)u[n]a^n$.

For an historical note, Robert G. Brown and Arthur D. Little used this method in 1956 in Exponential smoothing for predicting demand, apparently for the tobacco industry. A little more history and explanations can be found in Holt-Winters Forecasting for Dummies (or Developers) - Part I. Peter Zehna offers a critical review in Some remarks on exponential smoothing, 1966. A chapter by R. Brown in Encyclopedia of Operations Research and Management Science (Google books) dates the history back to 1944, the readable pages are reproduced here:

Many methods extend this smoothing, which lacks validity when the data has a trend or seasonality. Some of such are known as double or triple exponential smoothing, and Holt-Winters filters.

You can also check: How does this “simple filter” work?

Answer 3

Are there better approaches or further study on solutions to this which I should look at?

The normal approach for audio meters is a "lossy peak detector".

if new_value > current_value
  current_value = new_value;
else
 current_value = current_value * decay;  

This reacts immediately to any new or peak or transient in the signal but it lingers on for a while so it creates a much less hectic picture. Decay should be a constant between 0 and 1. It controls how quickly the bars come done with 0 being instantaneous and 1 being never.

Answer 4

Around US DoD contractor circles, this particular filter is frequently called an "alpha filter", because it can be characterized with one parameter that is traditionally named "alpha".

It is directly analogous to a simpe analog RC low-pass filter.

They are extremely simple, have serious limitations, but they have the undeniable advantage over more complex (and complicated!) filters that, if you steer clear of their problem areas, they get the job done.

Answer 5

As mentioned in other answers, this is a Single Pole Recursive filter, a type of infinite impulse response (IIR) filter.

A great resource for information on this and other DSP features is Steven W. Smith's The Scientist and Engineer's Guide to Digital Signal Processing:

...each point in the output signal is found by multiplying the values from the input signal by the "a" coefficients, multiplying the previously calculated values from the output signal by the "b" coefficients, and adding the products together.. Notice that there isn't a value for b0, because this corresponds to the sample being calculated. Equation 19-1 is called the recursion equation, and filters that use it are called recursive filters. The "a" and "b" values that define the filter are called the recursion coefficients.