It sounds like this“Turning a signal into its frequencies loses detail, like blurring a photo.”
Thinking a spectrum throws information away
Why it feels right
A summary usually loses detail. An average hides the spread, and a list of ingredients doesn't tell you how to bake. When you hear 'these are the frequencies in the sound', it sounds like a description of what's in it, not the sound itself, and descriptions lose things.
Why it’s wrong
The Fourier transform loses nothing. The signal and its frequency recipe, with strengths and the timing offset of each sine, are two descriptions of the same thing, and an inverse transform rebuilds the original exactly. Only if you keep the strengths and throw away the offsets do you lose information.
A worked example
Take a signal sampled at N points. The transform produces N numbers, one for each frequency, each carrying a strength and an offset in time, called a phase. That is as many numbers as you started with, and the operation can be reversed. Adding up the sines with those strengths and phases gives back the original samples, exactly.
It is like writing the same quantity in two units: the information is the same, but different things are easy to see. In the frequency picture, you can see at a glance that a recording has a hum at 50 Hz. In the time picture, you can see when a clap happened.
Information is lost only when you throw some away. A spectrum plot usually shows just the strengths, which drops the phase and so can't rebuild the signal. MP3 deliberately drops frequencies the ear barely hears. The transform itself discards nothing.
How to catch yourself
- You assume you can't get the signal back from its spectrum.
- You think the spectrum is a summary.
- You confuse the transform with compression.
What to hold instead
The Fourier transform is reversible: the strengths and timing offsets of all the sine components rebuild the signal exactly. Detail is lost only when something is deliberately discarded, as when MP3 compresses.
Where you’ll meet it
- Reading about audio compression.
- Seeing why filtering can be done in the frequency picture.
- Wondering whether a spectrum plot contains the whole recording.
Questions people ask
- Does the Fourier transform lose information?
- No. It re-expresses the same signal as strengths and phases of sine waves, and the inverse transform returns the original exactly. Any loss comes from discarding parts, such as phases or weak frequencies.
- What is phase in a Fourier transform?
- The timing offset of each sine component: where in its cycle it starts. Strengths alone tell you how much of each frequency is present, and phases tell you how to line them up to rebuild the signal.