Modulation and Demodulation of Biomedical Signals

451

Sq(t) can be written as a Fourier series (Northrop, 2003):

		•		n+1	cos[(2n − 1)ωct]
	Sq(t) = 1	2 + (2 π) (−1)				(11.47A)
					(2n − 1)
		n=	1

¬

Sq(t) = 12 + (2π){cos(ωct)− (13)cos(3ωct)+ (15)cos(5ωct)−≡} (11.47B)

Now multiply the Fourier series by the terms of Equation 11.46:

yd (t) = 12 A cos(ωct)+ (2π A{cos2 (ωct)− (13)cos(ωct)cos(3ωct)+

+(15)cos(ωct)cos(5ωct)−≡}+ 12 A mo cos(ωmt)cos(ωct)

+(2A mo π)cos(ωmt)cos2 (ωct)− (2A mo 3π)cos(ωmt)cos(ωct)cos(3ωct)

+(2A mo 5π)cos(ωmt)cos(ωct)cos(5ωct)

− (2A mo 7π)cos(ωmt)cos(ωct)cos(7ωct)+≡

(11.48)

and examine what happens when the terms of Equation 11.48 are passed through a band-pass filter that attenuates to zero dc and all terms at above (ωc − ωm). Trigonometric expansions of the form cos(x) cos(y) = (1/2) [cos(x + y) + cos(x − y)] are used. Let the BPF’s output be ymdf (t):

ymdf (t) = (A π)+ (A π)[mo cos(ωmt)]

(11.49)

The BPF output contains a dc term plus a term proportional to the desired mo cos(ωmt). Because AM radio is usually used to transmit audio signals that do not extend to zero frequency, the band-pass filter blocks the dc but passes modulating signal frequencies. Thus, ymdf (t) mo cos(ωmt). Several other AM detection schemes exist, including peak envelope detection and synchronous detection, described later in the detection of DSBSCM signals; the interested reader can find a good description of these modes of AM detection in Clarke and Hess (1971).

11.5.3Detection of FM Signals

When modulating and transmitting signals with a dc component, FM is the desired modulation scheme because a dc signal, Vm, produces a fixed frequency deviation from the carrier at ωc given by:

© 2004 by CRC Press LLC

452	Analysis and Application of Analog Electronic Circuits
	Δω = Kf Vm	(11.50)

As in the case of AM, FM demodulation can accomplished by several means. The first step in any FM demodulation scheme is to limit the received signal. Mathematically, limiting can be represented as passing the sinusoidal FM ym(t) through a signum function (symmetrical clipper); the clipper output is a square wave of peak height, ymcl(t) = B sgn[ym(t)]. (The sgn(ym) function is 1 for ym ≥ 0, and −1 for ym < 0.) Clipping removes most unwanted amplitude modulation, including noise on the received ym(t); this is one reason why FM radio is free of noise compared to AM. The frequency argument of ymcl(t) is the same as for the FM sinusoidal carrier, i.e., ωFM = ωc + Kf vm(t).

Once limited, several means of FM demodulation are now available, including the phase-shift discriminator; the Foster–Seely discriminator; the ratio detector; pulse averaging; and certain phase-locked loop circuits (Chirlian, 1981; Northrop, 1990). It is beyond the scope of this text to describe all of these FM demodulation circuits in detail, so the simple pulse averaging discriminator will first be examined.

In this FM demodulation means, the limited signal is fed into a one-shot multivibrator that triggers on the rising edge of each cycle of ymcl(t), producing a train of standard TTL pulses, each of fixed width δ = π/ωc sec. For simplicity, assume the peak height of each pulse is 5 V and low is 0 V. Now the average pulse voltage is vav(t):

	1	δ	ω	+ K	v	m	π ωc
vav (t) =		5 dt =		c f				5 dt = (5 2)(1+ Kf vm ωc )	(11.51)
	T			π
0							0

Thus, recovery of vm(t), even a dc vm, requires the linear operation:

vm (t) = [(2 5)vav (t) − 1](ωc Kf )

(11.52)

In practice, the averaging is done by a low-pass filter with break frequency

ωmmax ωf ωc.

In phase modulation, the modulated carrier is given by:

ym (t) = A cos[ωct + Kpvm (t)]

(11.53)

Because the frequency of the PhM carrier is the derivative of its phase,

∞	(11.54)
ωPhM = ωc + Kp vm (t)

PhM carriers can be generated and demodulated using phase-locked loops (Northrop, 1989).

© 2004 by CRC Press LLC

Modulation and Demodulation of Biomedical Signals

453

(FM process)

(PhD)

(Loop filter)

θi

vm(t)

vp

vc

Km

KP

Kf

s

(s + a)

− θo

(VCO)

Kv

s

FIGURE 11.12

A PLL used to demodulate an NBFM carrier.

Figure 11.12 illustrates an example of a PLL used to recover the modulating signal, vm(t), in an NBFM carrier. Note that the PLL input is the phase of the NBFM signal, θi, which is proportional to the integral of vm(t). The PLL tries to track θi(t) and in doing so generates the VCO control signal, vc(t), that is of interest. The frequency in the transfer function of the PLL is the frequency of vm(t), not ωc. The loop gain of the PLL is:

	A (s) = −	KPKf Kv	(11.55)
		s(s + b)
	L

To give the closed-loop PLL a damping of ξ = 0.707, it is easy to show, using root locus, that KP Kf Kv = b2/2 and the undamped natural frequency of the PLL is ωn = b2/2 r/s. The frequency response function for the PLL demodulator can be shown to be:

V	(jω) =	(Km jω)Kp	Kf (jω + b)			K K	(11.56)
Vm		1+ KPKf Kv	[jω(jω + b)] = (jω)2			(b2 2)+ jω(2 b)+ 1
c						m v

Thus, at signal frequencies below the loop’s ωn = b 2/2 r/s, the PLL demodulates the NBFM input; the output is proportional to vm(t). That is:

vc (t) (Km Kv )vm (t)

(11.57)

Therefore, the loop filter output is proportional to vm(t) for vm frequencies between zero and about ωn/2.

11.5.4Demodulation of DSBSCM Signals

The demodulation of DSBSCM carriers is generally done by a phase-sensitive rectifier (PSR) (also known as a synchronous rectifier) followed by a lowpass filter. One version of this system is shown in Figure 11.13. In this op amp

© 2004 by CRC Press LLC

454			Analysis and Application of Analog Electronic Circuits
					C
		R	R	R	R
		R			R
					___
		OA1	R/2		vz(t)
	DSBSC		−ym(t) MOS switch	OA2	OA3
Vo’					vz(t)
	signal

LPF

vr(t)

SYNC

FIGURE 11.13

Schematic of a three-op amp synchronous (phase-sensitive) rectifier used to demodulate a DSBSC-modulated carrier.

version of a PSR, an analog MOS switch is made to close for the positive half-cycles of the reference signal that has the same frequency and phase as the unmodulated carrier. Figure 11.14 illustrates a low-frequency modulating signal, vm(t), from there, the DSBSCM signal, and, finally, Vz(t), the unfiltered output of the PSR. Note that “c” means the MOS switch is closed and “o” means it is open. The simple op amp low-pass filter also inverts, so its output Vz is proportional to −vm(t).

Another means of demodulating a DSBSC signal is by an analog multiplier followed by an LPF. In the latter means, the multiplier output voltage, Vo′(t), is the product of a reference carrier and the DSBSCM signal:

Vo′(t) = (0.1)Bcos(ωct) ∞ (A mo2)[cos((ωc + ωm )t)+ cos((ωc − ωm )t)] (11.58)

The 0.1 constant is inherent to all analog multipliers. By trigonometric identity, noting that cosθ is an even function, the multiplier output can be written:

V′(t) = (0.1)(A m B 4) cos((2ω			c	+ ω	m	)t)+ cos(ω	m	t)	(11.59)
o	o	[						]

After unity-gain low-pass filtering,

	(t) = (0.1)(AB 4)mo cos(ωmt)	(11.60)
Vo′

which is certainly proportional to vm(t).

Still another way to demodulate DSBSC signals is by a special PLL architecture called the Costas loop (Northrop, 1990), shown in Figure 11.15. Its successful operation requires that the modulating signal, vm(t), be nonzero

© 2004 by CRC Press LLC

Modulation and Demodulation of Biomedical Signals

455

vm(t)

t

0

DSBSCM signal

0

	Vz	PSD signal
		(before LPF)
								Switch status
0	c	o	c	o	c	o	c	o	c	o

FIGURE 11.14

Waveforms relevant to the operation of the synchronous rectifier of Figure 11.13. Top: Modulating signal. Middle: DSBSC signal. Bottom: Detected signal before LPF.

only for short intervals so that the PLL does not lose lock. The input DSBSCM signal can be written:

x1 = vm(t) Vc cos(ωct + θc)

(11.61)

The output of the PLL’s VCO is:

x6 = X6 cos(ωot + θo)

(11.62)

© 2004 by CRC Press LLC

456	Analysis and Application of Analog Electronic Circuits
	M1
x1	x2	x3
		LPF1

x6 Kv	x5	LPF3	x4	M3
s

90o PS

x7

x8	x9
	LPF2

M2

FIGURE 11.15

Block diagram of a simple Costas PLL.

and the output of mixer M1 is:
x2 = x1 ∞ x6 = vm (t)VcX6 cos(ωct + θc )cos(ωot + θo )	(11.63)

= [vm (t)Vc X6 2]{cos[(ωc + ωo )t + θc + θo ]+ cos[(ωc − ωo )t + θc − θo ]}

At lock, ωo ωc and θo θc, and LPF1 passes only the low-frequency components of x2. Thus:

x3 = vm(t) Vc X6/2

(11.64)

which is the desired output.

Now examine how the other signals in the Costas loop contribute to its operation. By trigonometric identity, the output of the quadrature phase shifter is:

x7 = −X6 sin(ωot + θo)	(11.65)
The output of the second mixer is thus:
x8 = −X6vm (t)Vc sin(ωot + θo )cos(ωct + θc )	(11.66A)

x8 = [−X6vm (t)Vc 2]{sin[(ω0 + ωc )t + θo + θc ]+ sin[(ωo − ωc )t + θo − θc ]} (11.66B)

© 2004 by CRC Press LLC