What does Fast and Slow Time Weighting signify for real-time audio signal processing?

Question

I have a streaming audio signal that I am recording in real time. To make the explanation easy, say the signal that I receive is calibrated and has a sampling frequency, $F_s = 44100\textrm{ Hz}$.

I am then finding the Sound Pressure Level of this signal. For this purpose, I get a frame of audio data of t seconds.

Then, RMS Sound Pressure Level (dB) is given by: $$ \rm Amplitude_{RMS} = \sqrt{mean\left(frame ^2\right)} $$

This gives the instantaneous RMS of a $t$-second frame of audio. Then, to find a $\textrm{ dB}$ level, $$ L_p = 20\log_{10}\left(\rm Amplitude_{RMS}\right) $$

Things get tricky here with the definitions of 'Fast' or 'Slow' sound pressure level. As this post points out, for time-weighted exponential average, I created a low-pass filter with a real pole at $1/\tau$ (where $\tau= 125\textrm{ ms}$ for 'fast' or $1\textrm{ s}$ for 'slow')

τ = 0.125; %fast
[b a] = bilinear(1, [1 1/τ], Fs);
frame_filtered = filter(b, a, frame.^2);
L_fast = 20*log10(sqrt(mean(frame_filtered)));

The IEC 61672 specifies the time weightings to be Fast and Slow for determining the speed at which the instrument responds to changing noise levels.

• How does this work for digital audio?
• Does it mean that the analysis needs to be performed on a fixed frame of data?

• For this particular example, this frame would be $t\cdot F_s$ samples?

  Slow: 1*44100 = 441000 samples
  Fast: 0.125*44100 ~= 5513 samples

• If so, is it true that I would need to know in advance whether I need fast or slow levels?

Answer 1

By "time weighting" in acoustics we are applying an RC (resistor–capacitor) circuit to a time signal.

See: RC circuit on Wikipedia

The time constant for FAST is 0,125 s and 1 s for SLOW.

You can also apply this digitally on your time signal by using my code below - it is basically a exponential moving average filter. I think you understand the steps by just looking at the comments in the code below.

public class FilterTimeWeighting {

private int nBands = 29; //number of bands
private float initValue;

FilterTimeWeighting() {
    initValue=0;
}

/**
 * Apply the LAmax RC-filter on the signal. Valid for fsamp=44100 Hz.
 * Returns the signal in Volts after applying filter.
 * 
 * //y(n)=a*y(n-1)+(1-a)*x(n)
 *
 * 
 *
 * @param signalIn
 * @return
 */
public float[] applyLmaxFilter(float[] signal) {

    int n_samples = signal.length;

    float[] x_signal = new float[n_samples];
    float[] y_signal = new float[n_samples];
    float[] ret_signal = new float[n_samples];


    //Square the signal
    for (int i_sample = 0; i_sample < n_samples; i_sample++) {
        x_signal[i_sample] = signal[i_sample] * signal[i_sample];
    }

    //Exponential moving average parameters
    float tau=0.125f;
    float fs=44100f;
    float alpha=tau/(1f/fs+tau);

    //for the first sample 
    y_signal[0] = initValue*alpha+(1-alpha)*x_signal[0];

    //for the rest of the samples
    for (int i_sample = 1; i_sample < n_samples; i_sample++) {
        y_signal[i_sample] = y_signal[i_sample-1]*alpha+(1-alpha)*x_signal[i_sample];
    }

    //remember last sample for next time step;
    initValue=y_signal[n_samples-1];

    //Square root
    for (int i_sample = 0; i_sample < n_samples; i_sample++) {
        ret_signal[i_sample] = (float) Math.sqrt(y_signal[i_sample]);
    } 

    return ret_signal;
}

}