FPGA Central

I have a very basic conceptual question that I have been having trouble
getting my head around. When shifting the frequency of a sin wave, how
should the phase be handled? For example, this works for integer shift
amounts:

y(t) = sin(shift*(2*pi*f*t + theta))

But if shift is not an integer (1.2, 1.5, etc.) the phase of the
resulting sin wave seems incorrect.

Anybody know how phase should be multiplied when shifting frequency?
This is likely printed in several basic physics texts, but I haven't
stumbled upon it in the ones I have in front of me.

Thanks!
Bob

BobM said the following on 31/03/2006 02:22:
> I have a very basic conceptual question that I have been having trouble
> getting my head around. When shifting the frequency of a sin wave, how
> should the phase be handled? For example, this works for integer shift
> amounts:
>
> y(t) = sin(shift*(2*pi*f*t + theta))
>
> But if shift is not an integer (1.2, 1.5, etc.) the phase of the
> resulting sin wave seems incorrect.
>
> Anybody know how phase should be multiplied when shifting frequency?
> This is likely printed in several basic physics texts, but I haven't
> stumbled upon it in the ones I have in front of me.
>

Depends on what you want to happen to the phase (theta). If you're
treating it as an angular offset, and want to retain that offset at the
new frequency, then you should do:

y = sin(shift*2*pi*f*t + theta)

If, instead, you're treating it as the result of some kind of constant
time offset, then your original expression was correct (it retains the
same "time" offset at the new frequency).


--
Oli

"BobM" <[email protected]> wrote in message
news:[email protected] oups.com...
>I have a very basic conceptual question that I have been having trouble
> getting my head around. When shifting the frequency of a sin wave, how
> should the phase be handled? For example, this works for integer shift
> amounts:
>
> y(t) = sin(shift*(2*pi*f*t + theta))
>
> But if shift is not an integer (1.2, 1.5, etc.) the phase of the
> resulting sin wave seems incorrect.
>
> Anybody know how phase should be multiplied when shifting frequency?
> This is likely printed in several basic physics texts, but I haven't
> stumbled upon it in the ones I have in front of me.
>
> Thanks!
> Bob
>

Hello Bob,

It depends on how you implement the frequency shifting.

Let's say you have a sinusoid:

y=cos(at+p) where a is the frequency and p is the phase. Now let's operate
on this and shift if by frequency b. One way is to analytically extend your
function y(t) to now be

Y(t) = cos(at+p) + j*sin(at+p)

next mix with a complex exponential at frequency b.

h(t) = cos(bt)+j*sin(bt)

The frequency shifted portion of your original signal is the real part of
Y(t)*h(t)

thus:

Re(Y(t)*h(t)) = cos(at+p)*cos(bt) - sin(at+p)*sin(bt)

= cos( at + p +bt)

= cos ( (a+b)t + p )

Hence the result is shifted to a new frequency but with the same initial
phase!

IHTH,

Clay S. Turner