[BC] FM Stereo invention

[Robert Orban]

At 04:52 PM 2/19/2007, you wrote:
>In a message dated 02/19/2007 2:34:02 PM Eastern Standard Time,
>padrino at telos-systems.com writes:
>
> > What you're talking about is a simple sampling method. Not much different
> >  that what an A/D converter is doing at a base sampling rate. The theory is
> >  the same.
> >
> >  -Frank Foti
>
>Not exactly, tho... in a true sampling environment, the signal level is
>*held* until the next sample comes along. With the chopped 50/50 mpx, you 
>have
>alternating levels. With a stereo signal with a large variation between 
>channels,
>you'd have the level switching back and forth with a nearly 50% duty cycle.
>If, let's say, the L channel was silent, then you'd have the R being chopped
>on/off at that 50% duty cycle. It could be somewhat restored by a 
>sample-and-hold
>ckt in the receiver, but that's starting to get more complex than practical.
>A low-pass filter would only tend to make the chopping less pronounced, 
>but it
>would still be there. Making the filter too aggressive would sharply
>attenuate the highs.

Here is the derivation of the encoding equation for the switching 
generator. 'ws' is the sampling frequency. We multiply the left channel by 
a squarewave whose frequency is ws and which has maximum value is 2 and a 
minimum value of zero. We multiply the right channel by the same 
squarewave, but delayed by 180 degrees with reference to ws.

 From the Fourier transform of a square wave, we use only the DC term and 
the fundamental frequency, COS(ws), in the equation. To justify this, we 
know from AM modulation theory that the multiplication will produce 
symmetrical sidebands above and below the sampling frequency. Because the 
second harmonic of ws is at 76 kHz, the lowest sideband of this harmonic 
(assuming 15 kHz audio bandwidth) is 76 - 15 kHz = 61 kHz, so we can filter 
this out. Hence, we can ignore the higher harmonics of the square wave 
because the baseband lowpass filter will remove the spectrum around the 
harmonics. This is why I will only include the DC and fundamental terms in 
the equation.

The DC term of our squarewave is 1 and the peak magnitude of the 
fundamental is 4/pi. Because the peak magnitude of the fundamental is not 
the same as the DC term, we need to add a bit of extra L+R into the output 
of the switches to get the ideal FM matrix stereo waveform. Here is the 
equation, including the L+R correction term:

L[1+(4/pi)COS(ws)]+R[1-(4/pi)COS(ws)] + [(4-pi)/pi](L+R)

A little algebra shows that this equals

(4/pi)[(L-R)COS(ws)+L+R]

One can now clearly see the matrix representation, with L+R at baseband and 
L-R as a double sideband suppressed carrier AM signal generated by 
multiplying L-R by COS(ws).

(The 4/pi is a trivial gain factor).

Please note that these equations do not assume a sample-and-hold operation. 
Such an operation is unnecessary.



Bob Orban