# Peak detector circuit for AM demodulation

In Lee's Design of CMOS Radio Frequency Circuits, the leaky peak detector circuit for AM demodulation is discussed. The AM expression is:

$$v(t)=(1+mf(t))cos(\omega_ct)$$

And the associated frequency domain representation of the modulated baseband signal and the peak detector circuit are below:

1. As shown in figure 2.12, the modulated signal has impulses at the carrier frequency, which I understand. The book claims: "The carrier’s presence is not entirely without value, however, for it permits the use of a very unsophisticated demodulator [the leaky peak detector]." Why would this be the case? I'm not sure why the circuit of Fig 2.13 couldn't be used if the carrier were suppressed, i.e. if there were no constant 1 in the AM equation above. On the contrary it would seem like the presence of the carrier makes it harder to detect.

2. The book claims that "Furthermore, the RC product must be large enough to provide adequate filtering of the carrier “teeth” yet small enough to allow the tracking of the steepest portions of the envelope", and as a result says the following inequality needs to be satisfied:

$$\frac{1}{f(t)}\vert\frac{d}{dt}f(t)\vert<\frac{1}{RC}$$

The verbal explanation of the filter's bandwidth requirement makes sense to me but I'm not sure where this inequality comes from. Could someone point me in the direction to derive it?

• What do you mean by impulses at the carrier frequency? Have you looked at the waveform of an AM wave with the offset and then looked at it without the offset? It should be clear if you do. Jan 17, 2023 at 22:53
• @Andyaka Hi Andy, by impulses at the carrier frequency I mean the frequency domain dirac impulses at +/- w_c in the AM's signal fourier transform, as seen on the RHS of Fig 2.12. And yes I looked at such waveforms with and without the offset, for example here: imgur.com/a/Jn9KY15, where a sinc^2 pulse amplitude modulates a cosine carrier with (top) and without (bottom) the offset. Visually the bottom (without offset) seems to have a much clearer envelope relating to the original signal. Jan 17, 2023 at 23:14
• Try this: upload.wikimedia.org/wikipedia/commons/thumb/b/b0/… <-- the top two can be demodulated with a diode detector but the bottom one would have severe distortion. Jan 17, 2023 at 23:44