Quote:

Originally Posted by

**Soulburner**
That's why Dirac's claims are curious to me. I don't think you can alter impulse without changing the FR, and vise versa.

This is actually a pretty interesting question. What's often loosely called the "frequency response" is the Fourier transform of the impulse response. One property of the Fourier transform is

*uniqueness* - if you know the "frequency response", you know the impulse response, and vice versa. So, given this uniqueness property of the Fourier transform, how can you change the impulse response without changing the "frequency response?" You can't.

But this misses something. For physical systems, the impulse response is a real-valued function of time. That is, for any value of time

*t* that you pick, the impulse response at that time

*t* is just a real number. But the "frequency response", that is, the Fourier transform of the impulse response, is a

*complex-valued* function of frequency. This means that for each frequency

*f* that you choose, the "frequency response" is a complex number.

This complex number can be interpreted much like an ordered pair of numbers { x0, y0 } in an x-y plane (Cartesian coordinates). You can also think of such an x-y pair in terms of polar coordinates by picturing a line segment drawn from the origin to the point { x0, y0 }. This line segment has a length, or magnitude, and an angle. The magnitude is easily calculated from x0 and y0 using the Pythagorean theorem, and the angle can be calculated using an arctangent formula. In an analogous way, a complex number can be thought of as having a real and imaginary part (like the Cartesian coordinates of an x, y pair), or as a magnitude and angle. So at each frequency, the "frequency response" has a complex value, which can be expressed as

*a* + j

*b*, where a and b are real numbers, and j is the imaginary number sqrt(-1). Now, the frequency response

*magnitude* in dB is 20 * log10 (sqrt(a*a + b*b)) by using the Pythagorean theorem and the definition of dB. But the magnitude is only half the story! There is also the phase to be considered. Since the "frequency response", that is, the Fourier transform of the impulse response, is a complex-valued function, if you alter its phase vs. frequency without changing its magnitude, you have nonetheless changed it, and you have therefore changed the impulse response too.

So when we loosely say, "FR", this could be stated as "20 times the base 10 log of the

*magnitude* of the Fourier transform of the impulse response." But again, the magnitude of a frequency response is only half the story.

How do you alter the phase vs. frequency of a "frequency response" without changing its magnitude? With an all-pass filter. An all-pass filter can be as simple as a single biquad, or can get as complicated as one likes.

I hope I didn't get too crazy with this explanation.