Using the above example:

a.) The sign is (-1) ^1 or -1.

b.) The mantissa is the fraction part in binary. In the above example, this is the mantissa: 10110000000000000000000

Converting the binary fraction to decimal:

*(1 x 2^-1) + (0 x 2^-2) + (1 x 2^-3) + (1 x 2^-4) + 0…0 = 0.6875*

c.) The exponent section is 01111100, the decimal equivalent is:

*(0 x 2^0) + (0 x 2^1) + (1 x 2^2) + (1 x 2^3) + (1 x 2^4) + (1 x 2^5) + (1 x 2^6) + (0 x 2^7) = 124*

Using the above values to compute the decimal equivalent with the provided formulas:

*(-1) x (1.6875) x [2^ (124-127)] = -0.2109375*

What will happen in the analog to digital conversion is opposite. Given the decimal analog equivalent, the recording software will compute the 32-bit float binary equivalent.

### Advantages of Recording/Mixing at 32-bit float bit depth over 24-bit

If you have utilized the maximum bit depth recording capability of your audio interface (such as a soundcard), you will be recording at 24-bits. The digitized signal only has 24-bit resolution; however adding extra 8-bits will provide additional headroom and dynamic range to your digital signal with the following benefits:

1.) 32-bit audio allows processing of signals above 0dBFS “internally” without clipping in the digital domain. It is because of increased headroom or dynamic range. “Internally” means within your DAW (digital audio workstation software).For example in Cubase manual; it recommends projects to be recorded in 32-bit float format for two obvious reasons:

a.) Prevent digital distortion and clipping during tracking.

b.) Cubase internal audio processing is in 32-bit float (like most DAW) and tracking in 32-bit float means that the recorded quality will be intact /untouched (because it is also in 32-bit float). If you are using below 32-bit float, possible audio degradation will occur.

2.) While you prevent high possibility of clipping with 32-bit float, you have the same signal to noise ratio (SNR) as the 24-bit audio.

*Max SNR = 6 x bit depth used during recording*

So a 16-bit recording has a maximum SNR of 6 x 16= 96dB. Below are the maximum SNR of 24-bit audio and 32-bit float resolution:

*Max SNR of 24-bit = 6 x 24 = 144dB*

*Max SNR of 32-bit float = 6 x 24 bits (removing those extra 8 bits) = 144dB.*

Take note that recording at 32-bit float will result to large file sizes for your recording track which can be a downside if you have limited hard disk drive spaces. The file size is computed using this formula:

*File size of your digital audio in bytes = [(sampling rate) x (bit depth) x (number of channels) x (seconds)] / 8*

So a digital recording of 10 seconds audio sampled at 32-bit float/96KHz stereo would have a file size of:

*File size = [(96000) x (32) x (2) x (10)]/8 = 7,680,000 bytes or approximately 7.7 MB*

With pure 24-bit recording, the file size would only be: 5.76MB, so a 32-bit float audio would be around 34% larger.

*Content last updated on July 5, 2012*