# FM Deemphasis

An analog deemphasis filter:

```              R
o------/\/\/\/---+----o
|
= C
|
---
```

Has this transfer function:

```              1             1
----          ---
RC          tau
H(s) = ---------- = ----------
1             1
s + ----      s + ---
RC           tau
```

And has its -3 dB response, due to the pole, at

```   |H(j w_c)|^2 = 1/2  =>  s = j w_c = j (1/(RC))
```

Historically, this corner frequency of analog audio deemphasis filters been specified by the RC time constant used, called tau. So w_c = 1/tau.

FWIW, for standard tau values, some standard analog components would be:

```   tau = 75 us = (50K)(1.5 nF) = (50 ohms)(1.5 uF)
tau = 50 us = (50K)(1.0 nF) = (50 ohms)(1.0 uF)
```

In specifying tau for this digital deemphasis filter, tau specifies the *digital* corner frequency, w_c, desired.

The digital deemphasis filter design below, uses the "bilinear transformation" method of designing digital filters:

1. Convert digital specifications into the analog domain, by prewarping digital frequency specifications into analog frequencies.

```w_a = (2/T)tan(wT/2)
```

2. Use an analog filter design technique to design the filter.

3. Use the bilinear transformation to convert the analog filter design to a digital filter design.

```H(z) = H(s)|
s = (2/T)(1-z^-1)/(1+z^-1)
```

```        w_ca         1          1 - (-1) z^-1
H(z) = ---- * ----------- * -----------------------
2 fs        -w_ca             -w_ca
1 - -----         1 + -----
2 fs              2 fs
1 - ----------- z^-1
-w_ca
1 - -----
2 fs
```

We use this design technique, because it is an easy way to obtain a filter design with the -6 dB/octave roll-off required of the deemphasis filter.

Jackson, Leland B., _Digital_Filters_and_Signal_Processing_Second_Edition_, Kluwer Academic Publishers, 1989, pp 201-212

Orfanidis, Sophocles J., _Introduction_to_Signal_Processing_, Prentice Hall, 1996, pp 573-583

## Parameters

Sample Rate
Sampling frequency in Hz
Tau
Time constant in seconds (75us in US, 50us in EUR)