Log response even harmonics

[Old Nick]

Log response even harmonics.

Hi, I am looking for a way of mathematically proving that a sine wave multiplied by a logarithm (sine wave modulated light - logarithmic pixel) will cause distortions in the sine wave (the distortions being only the even harmonics). I know I can do this with an FFT, but I'd like to find a more elegant proof, explaining how the logarithmic resonse introduces these even harmonics. Whereas an FFT just examines a waveform and returns it's contents.
My maths skills are sadly no up to this though.

Cheers,

Nick

 

[cedance]

do you mean distortion in the sine wave generators from logarithmic shaping schemes??? i dont understand what you mean by sine wave multiplied by a logarithm. it doesnt make sense to interpret it literally as log(a) * sin(t) where a is constant...

cedance.

 

[Old Nick]

do you mean distortion in the sine wave generators from logarithmic shaping schemes??? i dont understand what you mean by sine wave multiplied by a logarithm. it doesnt make sense to interpret it literally as log(a) * sin(t) where a is constant...

cedance.

sorry, I mean a log of a sine wave. ie, a sine wave which is distorted by a logarithmic response of an amplifier.
It appears as a multiplication in small signal as it is an amplifier with a logarithmic response, but appears linear over a small range with a transimpedance gain.

 

[Kral]

Old Nick,
As I understand the situation, you have an amplifier with a logarithmic transfer function. The input is biased to be somewhere in the middle of the transfer function, since the logarithm of a negative number does not exist. Ignoring the DC component of the output, the output with a sinusoidal input will be neither even nor odd. In other words, if x is the input, f(x) ≠ f(-x) and f(x) ≠ -f(-x). A signal of this nature has both even and odd harmonics.
.
If, on the other hand, you are running the sinusoidal input through an amplifier that has equal logarithmic-like responses for both negative and positive inputs, then the output will be odd ( f(x) = -f(-x) ). In other words, the positive output will be a mirror image of the negative output. Under these conditions, the output is distorted but symmetrical. Symmetrical waveforms have only odd harmonics.
Regards,
Kral

 

[Old Nick]

Cheers,

The pixel is detecting (sinewave)modulated light, and obviously there is no such thing as -ve light. THe amp converts the photocurrent into a Voltage, and has a logarithmic response.

The greater the modulation depth, the greater the distortion on the output, (flattening the top of the sinewave), giving an non-symetric output. I use a 4-phase lock in amp etc for the demodulation. This 4-phase algorithms do not cancel odd harmonics, but do cancle the evens. And I am 99.99% sure that there ar no odd harmonics being added as the do not show up when the outputs are demodulated, and the waveform is non symetrical which again suggests even harmonics. I would just like to be able to prove this mathematically, rather than just demonstrate it with plots from simulation/experiment.