Modulation and Demodulation of Biomedical Signals							447
	ˆ	•
								(11.36)
	vm (t) = τk {U(t − tk ) = U(t − tk+1)}
		k = 2
	where
	τk	∫ 1 τk	=		1	, k = 2, 3,	≡	(11.37)
				tk	− tk−1

Note that two pulses (k = 2) are required to define the first pulse interval. vˆm(t) is a series of steps and the height of each is the (previous) rk. Note

that the IPFD is not limited to the demodulation of IPFM; it has been experimentally applied as a descriptor to actual neural spike signals (Northrop, 2001).

11.5 Demodulation of Modulated Sinusoidal Carriers

11.5.1Introduction

Of equal importance in the discussion of modulation is the process of demodulation or detection, in which the modulating signal, vm(t), is recovered from the modulated carrier, ym(t). Generally, several circuit architectures can be used to modulate a given type of modulated signal and several ways can be used to demodulate it. In the case of AM (or single-sideband AM), the signal recovery process is called detection.

11.5.2Detection of AM

There are several practical means of AM detection (Clarke and Hess, 1971, Chapter 10). One simple form is to rectify and low-pass filter ym(t). Because rectification is difficult at high radio frequencies, heterodyning or mixing the incoming high-frequency AM signal with a local oscillator (in the receiver)

whose frequency is separated by a fixed interval,	f, from that of the incom-
ing signal produces mixing frequencies of (fc −	flo) = f. f is called the

intermediate frequency of the receiver and is typically 50 to 455 kHz, or 10.7 MHz in commercial FM receivers. High-frequency selectivity IF transformers are used that have tuned primary and secondary windings. Very high IF frequency selectivity can be obtained with piezoelectric crystal filters or surface acoustic wave (SAW) tuned IF filters.

Another form of AM detection passes ym(t) through a square-law nonlinearity followed by a low-pass filter; however, the square-law detector suffers

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448	Analysis and Application of Analog Electronic Circuits

			Audio BPF
Audio	x9		1	x8		M3	x5’
out			ω
							+90o	Phase
RF AM signal
								shift
Antenna			IF BPF					x5
								ωif
	M1
x1		xi	KI			ω4		IF
ωc, θc								XO
					x4
				ω			M2
Tuned			ωif
RF amplifier

xo

Kv

ωo, θo s

	LPF
x7	x6
	ω

FIGURE 11.10

Block diagram of a PLL system used to demodulate AM audio signals. Note that three mixers (multipliers) are used; M1 is the quadrature phase detector of the PLL.

from the disadvantage that it generates a second harmonic component of the recovered modulating signal. A third way to demodulate an AM carrier of the form given by Equation 11.1B is to mix (multiply) it by a sinusoidal signal of the same frequency and phase as the carrier component of the ym(t). A phase-locked loop system architecture can be used for this purpose; see Figure 11.10 (Northrop, 1990). This PLL system uses three multiplicative mixers. The AM input after RF amplification is:

x1 = A[1+ m(t)]sin(ωct + θc )

(11.38)

Mixer M1 effectively multiplies x1 ∞ xo. xo is the output of the PLL’s VCO. Thus,

xi = A[1+ m(t)]sin(ωct + θc ) ∞ Xo cos(ωot + θo )

(11.39)

= [A Xo [1+ m(t)]2]{sin[(ωc + ωo )t + θc + θo ]+ sin[(ωc − ωo )t + θc − θo ]}

The IF band-pass filter selects the second term, which has frequency close to ωif when the PLL is near lock. This means that x4 is:

x4 = [Ki A Xo [1+ m(t)] 2]sin[(ωc − ωo )t + θc − θo ]

(11.40)

Also, the output of the crystal-controlled oscillator (XO) is:
x5 = X5 cos(ωif t)	(11.41)

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Modulation and Demodulation of Biomedical Signals

449

Mathematically, the product, x6 = x4 ∞ x5 is formed:

x6 = [Ki A Xo [1+ m(t)]2]sin[(ωc − ωo )t + θc − θo ] ∞ X5 cos(ωif t) (11.42A)

x6 = X5[Ki A Xo [1+ m(t)]4]sin[(ωc − ωo + wif )+ θc − θo ]

(11.42B)

+ sin[(ωc − ωo − ωif )+ θc − θo ]

At lock, (ωc − ωo) = ωif r/s and (θc − θo) 0. The low-pass filter acts on x6; its output is the zero-frequency component of x6. Thus:

[	− ωo − ωif )+ θc − θo	]
X7 = X5[Ki A Xo [1+ m(t)] 4]sin (ωc			(11.43)

Note that x7 0 at lock. Now, x5′ = X5 sin(ωif t), so x8 can be written:

x8 = x4 ∞ x′5 = [Ki A Xo[1+ m(t)2]]sin[(ωc − ωo )t + θc − θo ]∞

(11.44A)

X5 sin(ωif t)

x8 = [X5Ki A Xo [1 + m(t) 4]]
cos (θc − θo − ωif )t + θc − θo			]	− cos (θc − θo + ωif )t + θc		− θo		]}	(11.44B)
{	[			[
Again, at lock:
x8 = [X5 Ki A Xo[1+ m(t)		{			[		]}
		4]] cos(0)			− cos (2ωif )t + θc − θo				(11.44C)

The band-pass filter cuts the dc term in x8 and also the 2ωif term; therefore, the output of the BPF is proportional to the modulating signal:

x9 = X5 Ki A Xo m(t) 4

(11.45)

Thus, the normalized modulating signal, [mo cos(ωm t)], is recovered, times a scaling constant.

A fourth kind of AM demodulation can be accomplished by finding the magnitude of the modulated signal’s analytical signal (Northrop, 2003).

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450	Analysis and Application of Analog Electronic Circuits

m(t)

A

ym(t)

B

+

ym(t)

HWR	BPF
ymr
ymr(t)	ymdf(t)
ym
	f

C

FIGURE 11.11

AM detection by simple half-wave rectification. (A) The modulating signal. (B) The AM carrier (drawn as a triangle wave instead of a sinusoid). (C) An ideal half-wave rectifier followed by an audio band-pass filter.

Examine the widely used rectifier + low-pass filter (average envelope) AM detector. Figure 11.11(A) illustrates a low-frequency modulating signal, m(t). Figure 11.11(B) shows the amplitude-modulated carrier (the sinusoidal carrier is drawn with straight lines for simplicity). Figure 11.11(C) illustrates the block diagram of a simple half-wave rectifier circuit followed by a bandpass filter to exclude dc and terms of carrier frequency and higher. The halfwave rectification process can be thought of as multiplying the AM ym(t) by a 0,1 switching function, Sq(t), in phase with the carrier. Mathematically, this can be stated as:

ymr (t) = A[1 + m cos(ωmt)]cos(ωct)Sq(t)

(11.46)

= A cos(ωct)Sq(t) + A mo cos(ωmt)cos(ωct)Sq(t)

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