Fast Fourier Transform to View Audio Frequency Spectrum

If you are dealing with frequency analysis of the audio wave “quantitatively”, one of the best tools I have learned in my signals/engineering class is FFT or Fast Fourier Transform. It is an algorithm to compute the Fourier transform equivalent of a time domain.

This highly advanced technique is very simple to understand, it simply converts a time domain function into a frequency domain function.

After audio mix down (where all sound of the instruments are cohesively combined into a single wave), the song is represented as a time domain function as we can see that the x – axis of the wave is using a time element in hours: minutes: seconds (only minutes and seconds is used realistically). See screen shot below:

But time domain graph of the audio wave specially used during the mastering process of the track is simply a plot of amplitude (y-axis) versus time. Obviously you cannot see the frequencies of that wave. It is why we used FFT (Fast Fourier Transform) to convert this time domain representation into a frequency domain plot. With frequency domain, you can analyze the amplitude (y-axis) versus Frequencies.

Though, it is highly recommended to stick to your ear when working with commercial audio production however you can see a glimpse of the frequency response of the track. For example, if you like to check if the uppermost frequency ranges (20,000Hz) has been already filtered. It is very hard to detect it using a human ear. In this case, you need a spectrum plot.

Using Adobe Audition, you can open your wave in edit view, and then click “Analyze” –> Show frequency analysis –> Set the FFT size to a maximum, as this increases the resolution and accuracy of the frequency plot. Set the type to default (Blackmann – Harris) and there you can see the frequency response graph. See below:

As you can see, it is confirmed that that a sharp low pass filter has been implemented with cutoff at 20,000 Hz (see plot above). This low pass filter can be implemented using Butter worth. (For beginners, read this.)

This is just an example of the wonderful application of FFT algorithm to see the Frequency spectrum. I won’t go into details as it becomes technically difficult to be understood by common readers; because of the extreme use of mathematics.

For students, please check this out:

Fast Fourier Transform Wiki notes
What is Fourier Transform?

Or if you are in school and your laboratory includes a frequency analyzer equipment; try running any time-domain signal into it (such as sine wave tone).Then confirm the graph and understand the relationships between a time-domain function to its frequency domain equivalent.

Most engineering students might perform this experiment. You also need an oscilloscope to view the waveform of a time domain signal. If you using some audio software with FFT capabilities (such as Audacity), you can see their differences below:

Time domain waveform of a 440Hz sine wave:

Frequency domain waveform of a 440Hz sine wave:

Best practices on using FFT Frequency Spectrum Analysis

1.) For best accuracy, always use the highest possible FFT size. In Adobe Audition 1.5, the highest possible is 65536.

2.) For best real time frequency domain representation (if you would like to see a real time plot of frequency vs amplitude when an audio is being played) you will need to set it at FFT size of 1024.

3.) If you have options in your FFT, you can start with the Blackmann-Harris algorithm.

4.) You can also use free software like Audacity as it includes a FFT frequency spectrum feature. In Windows 7 Audacity version 1.3 Beta, you simply need to go to Analyze — Plot spectrum. In there you can adjust the FFT settings, algorithm, etc.

5.) Never rely on FFT spectrum graph alone in your audio mixing and mastering activities. You need to use your ears and other tools to come up with the desired audio mixing settings. I recommend you read the following tutorials for additional information:

AudioSpectro FIRE– A tool for attaining frequency balance in audio mastering.

Basics on frequency analysis – a tutorial with emphasis on analyzing audio waveforms during mixing.

6.) Use FFT spectrum analysis to confirm EQ corrections and filtering. Although the ears can be a very accurate instrument in a highly precise monitoring environment; it is smart to double check the effect of EQ adjustments by examining the frequency spectrum.

Content last updated on July 4, 2012