Fourier Transforms, Phase Shift

Phase Shift

Use the angle addition formula to build a new fourier series. For instance, sin(3(x+p)) becomes cos(3p)×sin(3x) + sin(3p)×cos(3x). Make a similar change to cos(3(x+p)) and recombine terms. This substitution modifies a₃ and b₃, giving new coefficients c₃ and d₃. If we group c₃ and d₃ together, and evaluate at x, the result is the same as the original fourier functions a₃ and b₃ applied to x+p.

Modify the coefficients for all the multiples of x. Thus every a_i,b_i pair is adjusted according to the phase shift, giving a new c_i,d_i pair.

The new coefficients are linear combinations of the old coefficients, with multipliers that are bounded by 1. Use this fact to verify that the new coefficients approach 0. And if the coefficients approach 0, the terms approach 0, uniformly, for any x.

Let's see if the new series is convergent. Half of the partial sums, on the pair boundaries, are unchanged, and approach g(x+p). Since the individual terms approach 0, the new series converges to g(x+p). In fact, if the original series converges uniformly, then so does the new series.

What if we had derived the fourier series for g(x+p) directly, using integration? Would we generate the same series, with the same coefficients?

Consider the integral of g(x+p)×sin(x). Substitute x = t-p and get the integral of g(t)×sin(t-p). Referring to the original coefficients, this integral works out to be cos(p)×b₁ - sin(p)×a₁. This becomes the new coefficient c₁ on sin(x). Now, remember what happened when we adjusted the pair a₁cos(x+p) + b₁sin(x+p). sure enough, the new coefficient c₁ is the same.

Use analogous reasoning to build d₁ from the integral of g(x+p)×cos(x), and note that this is the same d₁ we obtained by phase shifting a₁ and b₁.

Generalize to the higher multiples of x. The fourier series of g(x+p) is precisely the shifted version of the fourier series of g.

This is no surprise. By assumption, integration on the fourier series of g extracts the a,b coefficients. This means the series is integrable term by term, and is (probably) uniformly convergent. The shifted series is uniformly convergent as well, and its c,d coefficients can be faithfully rederived using integration.

Here is the practical application of this theorem. If we want to find the fourier series for a given wave form, we can start the wave at any convenient offset, then, assuming the coefficients approach 0, we can phase shift it back into position later.

What about higher dimensions? If we can shift the wave form in one dimension, we can do it n times, for all n dimensions, and we're done. Shifting the series from x to x+p is accomplished using the method outlined above. It really isn't any different. For example, in one block of a 2 dimensional transform, the coefficients on sin(2x)×sin(3y) and cos(2x)×sin(3y) are adjusted, then, in the same block and in the same way, the coefficients on sin(2x)×cos(3y) and cos(2x)×cos(3y) are adjusted. Do this for each pair, in each block, and derive the phase shifted series. The "pairs" may not be adjacent, but they are always part of a fixed sized block, so the phase shift preserves convergence, and uniform convergence.

Shift the next coordinate, and the next, and translate the wave form as you wish.

Use a proof like the one shown above, and the coefficients produced by integrating g(x+p) are the same as the coefficients of g, phase shifted by p. The multiple integral of g becomes a nested integral, and we can start by integrating with respect to x. Replace x with t-p and expand sin(t-p), as we did before. The rest of the nested integral is unaffected, and c_i and d_i are expressed in terms of a_i and b_i. The relationship is the same as that produced byphase shifting.

Next, shift y by its offset, and shift z by its offset, and so on. At each step, integration and phase shifting produce the same fourier coefficients. After n steps the translation is complete.