PSD–autocorrelation relation; stereo FM

Relation between PSD and autocorrelation

By the Wiener–Khinchin theorem, the power spectral density (PSD) $S_X(f)$ of a wide-sense stationary process and its autocorrelation function $R_X(\tau)$ form a Fourier transform pair:

$$S_X(f) = \int_{-\infty}^{\infty}R_X(\tau)\,e^{-j2\pi f\tau}d\tau, \qquad R_X(\tau) = \int_{-\infty}^{\infty}S_X(f)\,e^{j2\pi f\tau}df$$

Consequences:

• Total average power: $P = R_X(0) = \int S_X(f)\,df$.
• $S_X(f)$ is real and even (because $R_X(\tau)$ is real and even), and it is never negative.
• Example: white noise $R(\tau) = \frac{N_0}{2}\delta(\tau) \Leftrightarrow S(f) = \frac{N_0}{2}$.

Stereo FM encoder

Stereo FM sends left (L) and right (R) channels on one carrier while staying compatible with mono receivers.

 L --+-->(+)--L+R-->[LPF]----------->( Σ )-->
     |    ^                            ^ ^  to FM
 R --|----+                            | |  mod.
     +-->(-)--L-R-->[LPF]-->[Bal.mod]--+ |
          ^ (R)                 ^ 38k    |
                  [19k osc]->[x2]        |
                      |                  |
                      +--19 kHz pilot----+

1. Matrix forms $L+R$ and $L-R$; both are band-limited to 15 kHz and pre-emphasized.
2. $L-R$ DSB-SC modulates a 38 kHz sub-carrier in a balanced modulator.
3. The 38 kHz sub-carrier comes from a 19 kHz oscillator and a doubler; a small 19 kHz pilot is also added.
4. Composite signal:

$$m(t) = (L+R) + (L-R)\cos(2\pi\,38\text{k}\,t) + A_p\cos(2\pi\,19\text{k}\,t)$$

Spectrum

 |  L+R      pilot    L-R DSB-SC
 |______       |    ____    ____
 |      |      |   |    |  |    |
 +------+------+---+----+--+----+---> f (kHz)
 0     15     19  23   38      53

Stereo FM decoder

FM    +->[LPF 0-15k]------------L+R--+->(+)--> 2L
demod |                              |   ^
out --+->[BPF 23-53k]-->( X )->[LPF]-+   |
      |                   ^      L-R  |  |
      |                   | 38k       +--|-->(-)--> 2R
      +->[BPF 19k]-->[ x2 ]              |
         (pilot)                  L-R ---+

1. FM discriminator gives the composite signal.
2. LPF (0–15 kHz) gives $L+R$ (all a mono set needs).
3. A narrow BPF picks the 19 kHz pilot; a doubler makes a phase-locked 38 kHz carrier.
4. BPF (23–53 kHz) selects the $L-R$ DSB-SC signal, which is coherently detected with the 38 kHz carrier and low-pass filtered.
5. Matrix: $(L+R)+(L-R) = 2L$ and $(L+R)-(L-R) = 2R$, then de-emphasis and audio amplifiers.