So when we want to reconstruct the signal again, that's just summation,

right, the basis elements times the inner product of the x's and the y's.

So we can do tricks, for example, if we want to reconstruct the signal,

but only omit, only include those terms that are very low frequency.

So for example, if we want this big rolling term but we want to get rid of

the high frequency information, we might do what is called a low pass filter.

So we would let the low-frequency terms through, and

we would filter out the high-frequency terms.

So this would amount to, just in our reconstructed signal here,

only adding those components associated with the low-frequency terms in the basis.

And conversely, we might want to just capture the high-frequency stuff and

filter out the low-frequency stuff.

So in this sum, we would just take the high-frequency terms and omit the others.

And then the reconstructed signal would have filtered out the low-frequency

information.

So again, this all boils down to least squares, like we've been studying.

However, I think it's fair to say that signal processors tend to

think of this in a different way.

But it's nice to put this very important concept, Fourier analysis,

in the domain of least squares, which we've been talking about in this class.

The second basis that I would mention to discuss a little bit is so-called

wavelet bases.

And wavelet bases are similar to Fourier bases.

They're an orthonormal basis that have some nice properties associated with them.

And there is a discrete wavelet transform,

just like there's a discrete Fourier transform.

But it's actually able to get the transform faster than if you were to do

it by calculating each of these inner products by themselves.