- Principle
- Mathematical description

- Digitization considerations
- MATLAB
^{TM}implementation - C implementation considerations
- Conclusion

Slope detector

(O.1) |

a differentiation corresponds to a filter having the transfer function
. The ampltitude plot of this filter is shown in figure O.1.
A plot of the signal before and after differentiation is shown in figure jtkr04:fig:slope02a and jtkr04:fig:slope02b. Obviously differentiation does not eliminate the frequencies around the carrier wave.

Differentiating the FM-signal and taking the absolute value produces a DC-offset and the amplitude varying input signal. This is shown in figure jtkr04:fig:slope02c. The DC offset and the frequency contents near the carrier that is still present, is removed by band pass filtering. The result is shown in figure jtkr04:fig:slope02d. The absolute value and low pass filtering corresponds to the envelope of the AM signal.

(O.2) |

where:

is the incoming FM signal. | [V] |

is the carrier frequency. | [rad/s] |

is the messages signal. | [V] |

is the carrier amplitude. | [rad/V] |

is the frequency sensitivity. | [Hz/V] |

By means of differention the FM signal, is transformed into an AM modulated FM signal, see figure O.2b. For simplicity is subsituted by c in the following and is omitted.

(O.3) | |||

(O.4) | |||

(O.5) |

Now the absolute value is taken, see figure O.2 c.

(O.6) | |||

(O.7) |

As a final step this new signal is bandpass filtered. This approach will only work if highest frequency of the wanted signal significantly lower than the carrier frequncy [Carlson, 1986, pages 259-261], see figure O.2 d. This yields

(O.8) |

where:

is a constant. | [-] |

That is - in the range of linearity the sum of the two transferfunctions will be constant. Furthermore at , and will be equal in amplitude. These proporties are shown in figure O.4a. The two signals at hand are: the envelope of the modulated signal passed through and the envelope of the modulated signal passed through . When these two are subtracted the offset amplitude cancels out and a frequency response of figure O.4b is achieved. Note that the resulting slope is twice as steep as either one of the slopes, or . The rule for the applicability of this method is the same as for the frequency discriminator.

[Haykin, 2001, pages 121-124,725-730]

Another positive attribute of the digital signal processing is the ability to produce high-order filters with relative ease.

Jes Toft Kristensen 2005-12-13