Performance Evaluation of the Hilbert Transform BasedDigital Phase-Locked Loop

Ivica Kopriva and Dragan JuricicInstitute for Defense Studies, R&D, Bijenicka 46, 10000 Zagreb, Croatia

e-mail: ivica.kopriva@public.srce.hr, dragan.juricic@public.srce.hr

Abstract

Multiplier free Digital Phase-LockedLoop (DPLL) based on Hilbert transform hasbeen analyzed in this paper. The performanceof this DPLL when applied as FM demodulatoris evaluated in relation to the DPLL withMultiplier based Phase Detector (MPD)showing better dynamic properties expressed inthe form of the lock range, settling time, pull-outrange etc.

1. Introduction

In the field of communication, thephase-locked loop (PLL) is frequently used forfrequency demodulation [1],[2]. In analog PLLand appropriate variant of the DPLL, theanalog and digital multipliers are used as PhaseDetectors (PD). The MPD measures the phasedifference between the input and output signalsand produces an error signal proportional to themeasured phase difference. The output signal ofthe MPD consists of a DC component (phasedifference) and an unwanted AC component(high frequency component). The Low Pass(LP) filter in the loop removes the ACcomponent of the MPD. The loop filter transferfunction has an important influence on thedynamic performance of the loop. If we want touse the PLL to demodulate frequencymodulation the loop filter must be wide enoughto obtain best loop tracking and acquisitionproperties, or the loop bandwidth should bemade as large as possible to minimize phase

error. Simultaneously, the loop filter shouldreject undesired high frequency component ofthe MPD and it must be chosen as LP filter. Itcan be shown that if cuttoff frequency of theloop filter is too low and the frequencydeviation of the input FM signal is excessive,the PLL can lose lock. Actually, there is aconflict in design of the loop filter and desireddynamic features of the PLL [3].

2. Hilbert transform PD

The Hilbert Transform based PhaseDetector (HTPD), which employs quadraturesignal processing method [4], estimates phasedifference between input and output signalswithout using LP filter:

Figure 1: HTPD operating principle

This PD extracts the phase error between theinput )(1 nu and output )(2 nu signals of thePLL by complex multiplication of complexsignals [5],[6],[7]. A complex signal of the form

)(~)( nujnu ±± is synthesized from a real signal)(nu and its Hilbert Transform and is known

as an analytic signal. The input and outputanalytic signals of the PLL are defined by:

X( . )+jH{ .}

( . )-jH{ .}

Im{ . }u1(n) ud(n)

u2(n)

[ ]

[ ])(2222

)(1111

20

10

)(ˆ)()(

)(ˆ)()(nnj

nnj

eAnujnunU

eAnujnunUϕω

ϕω

+

+

=−=

=+= (1)

The complex multiplication of these analyticsignals produces an output analytic signal of thePD. The real signal of the PD can be obtainedby taking imaginary part of this analytic signal:

[ ]{ }[ ] [ ])(sin)()(sin

)(

212121

)()(21

21

nAAnnAAeAAIMnu

e

nnjd

ϕϕϕ

ϕϕ

=−=== −

(2)

It can be seen that the output signal of theHTPD )(nud consists of a DC component and

does not contain the unwanted AC component.Consequently, there is no need for using a LPfilter as a PLL component building block. Thedigital controlled oscillator (DCO) used in thistype of PLL generates two signals, an in-phasesignal I and a quadrature signal Q. The DCO isnot able to compute I and Q signals directly.These signal are calculated indirectly from thephase )(2 nφφ . I signal and Q signals are theHilbert-transform pair. The signal flow diagramfor this DPLL is illustrated in Fig. 2.

F I R H T x

x

-1

D C O

+u

1(n)

I Q

2

1−−

M

z

Waveformsynthesizer θ2(n) 1

0

−z

TK

ud(n)

PD

Figure 2. The digital PLL with HTPD

As indicated in the block diagram, this type ofDPLL is a first-order loop because the loopfilter is omitted. The loop delays input signal by(M-1)/2 time steps and the output signal ofHTPD directly controls the DCO.

3. Performance analysis

Performance of the DPLL with HTPDis compared with the DPLL with MPD throughevaluation of the four key parameters specifyingfrequency range in which PLL can be operated.The loop filter has an important influence onthese parameters. They are defined for bothvariants of the loop as:

Digital PLL withLP filter

DigitalPLL with

HTPD

The hold range

( Hωω∆∆ )∞→)0(HKK do do KK

The lock range

( Lωω∆∆ )ξω

τ

τn

do KK2

1

2 = do KK

The pull-outrange

( POωω∆∆ )

)1(8.1 +≈ ξω n do KK

The pull-in

range ( Pωω∆∆ )∞→

do KK

where nωω is the loop natural frequency, dK is

gain of PD and 0K is gain of the VCO. If, the

PLL is pulled-out by a large frequency step, wecan expect that PLL to come back to stableoperation. This process is known as pull-inprocess and is relatively slow. The frequencystep which causes loop to lose lock, is muchgreater for the PLL with HTPD than the PLLwith MPD. The DPLL with HTPD has largerlock range than a DPLL with MPD and is ableto track faster phase and frequency variationsof the input signal. Moreover, the loopbandwidth for the DPLL with HTPD is greaterthan the DPLL with MPD as follow:

The loopbandwidth

DPLL with LPfilter

DPLLwith

HTPD

gωω1ττ

ωωωω dong

KK==≈≈dog KK≈≈ωω

It is clear that the loop filter reduces the loopbandwidth. Consequently, the lock-in time(settling time) for the PLL with HTPD is veryshort.

One of the main advantages of using thePLL is its ability to demodulate and track signalin noise. The PLL offers excellent noiseimmunity compared to other designs [8],[9].When noise is superimposed on input signalloop bandwidth must be as narrow as possibleto minimize output phase jitter. Whereas, theloop bandwidth should be made as large aspossible to obtain best tracking and acquisitionproperties [1]. Than, there is a problem whenwe try to choose appropriate parameters of theloop. The output signal-to-noise ratio outSNR

versus the input signal-to-noise ratio inSNR

curve characterizes the PLL FM demodulatorperformance in the presence of noise. Tocalculate the signal-to-noise ratio of the VCOoutput outSNR , it is necessary to know the

inSNR , the noise bandwidth of the loop LB

and the input noise bandwidth iB as follow [3]:

inL

iout SNR

BB

SNR == (3)

If the input bandpass filter is considered as apart of the PLL FM demodulator the noisebandwidth iB is the bandwidth of such a filter.

The noise bandwidth of the loop for both typesof the PLL design is defined as [3],[10]:

Digital PLL withLP filter

Digital PLLwith HTPD

The loopbandwidth

[Hz ]

++==

ζζζζ

ωω41

2n

LB4

odL

KKB ==

The PLL with HTPD has larger noisebandwidth of the loop than digital PLL withmultiplier PD. Thus, this PLL has less rejectionof the noise superimposed on input signal. In

practice, when PLL is used as FMdemodulator we must filter the output signal forboth variants of the PLL design. Therectangular passband postfilter with cuttofffrequency equal to the massage bandwidthshould be used [9]. Therefore, thesedemodulators have approximately equal noiseperformance when the postfilter is used.

4. Simulation results

In this paper, we simulated behavior ofthe digital PLL with the MPD and digital PLLwith HTPD. The loops are implemented bysoftware on DSP TMS320C40 (TI) in realtime [11]. First loop (PLL with MPD) is theordinary second–order loop with the activeloop filter and flat amplitude response

70.0==ζζ (damping factor). The special 80th-order FIR filter is used as Hilbert transformer inthe second loop. The loops are designed inorder to demodulate a real FM signal whichhas the center frequency kHzfC 5.22== , the

frequency deviation kHzf 10==∆∆ , themodulation frequency Hzfm 75== and

amplitude VA 11 == . The sampling frequency is100 kHz.

∆+= )2sin(2sin)( 11 tf

ff

tfAtu mm

c ππ (4)

Figure 3. The spectrum of the input signal

frequency [ kHz]

mag [db]

Figure 4. The spectrum of demodulated signal (the PLL with MPD)

time [s]

voltage [V]

Figure 5. The demodulated signal

frequency [ kHz]

mag [ db]

Figure 6. The spectrum of demodulated signal (the PLL with HTPD)

time [s]

voltage [V]

Figure 7. The demodulated signal

As illustrated (Fig.6), the output signal of thePLL with HTPD does not contain the unwanted

AC component near frequency cf2 , contrary

to the digital PLL with MPD (Fig. 4). When wewant to design the digital PLL with MPD wemust choose the parameters ζζ and nωω , which

determine the frequency response and thephase error of the loop. Whereas, the phaseerror depends on frequency deviation f∆∆ andmodulation frequency mf of the input FM signal

[1]. The phase error must be small and liewithin the linear range of the loop all the time. Inthat case, demodulating signal is reproduced atthe loop output with minimum distortion [8].For greater modulation frequency which isequal to Hzfm 750== but still smaller than the

loop bandwidth, this PLL will lose lock (Fig. 8and 9). The PLL with HTPD has larger loopbandwidth and can track the new modulationfrequency of the input signal. Generally, there isno tracking problem for this PLL when theinput frequency lies in the hold range of the loop(Fig. 10 and 11).

frequency [kHz]

mag [ db]

Figure 8. The spectum of demodulated signal (the PLL with MPD)

time [s]

voltage [V]

Figure 9. The demodulated signal

frequency [ kHz]

mag [ db]

Figure 10. The spectrum of demodulated signal (the PLL with HTPD)

time [s]

voltage [V]

Figure 11. The demodulated signal

5. Conclusions

The design of digital PLL as FMdemodulator with HTPD and the main results ofthe performance analysis are presented in thispaper. The computational operations requiredfor this PLL suggest implementation bysoftware. The key element of this PLL is theHTPD which extracts the phase error bymathematical computations without using a LPfilter. Consequently, this PLL has largerbandwidth, pull-out and lock range than digitalPLL with MPD. Therefore, it is able to trackfaster frequency variations of the input signalwhich instantaneous frequency lies in the holdrange of the loop. This PLL has less rejectionof the noise superimposed on the input signaldue to the larger noise bandwidth of the loop.The output signal of the loop is delayed by (M-1)/2 time steps because the ideal Hilberttransformer is realized as FIR structure [6],[7].The sampling frequency must be at least twicethe highest-frequency component existing in

input signal [12].That is not case for the PLLwith MPD where the sampling frequency mustbe at least four time greater than the highest-frequency component existing in input signal.Whereas, the software algorithm for this PLL ismore complicated than algorithm of the PLLwith MPD.

References

[1] B.T. Mills, “The Phase Locked Loop as a Communication System Building Bock”,NS Application Note 46, June 1971.[2] H.Taub, D.L. Schilling, “Principles ofCommunication Systems”, McGRaw-Hill,1987.[3] R. Best, “PLL-Theory, Design andApplications”, McGraw-Hill, 1993.[4] P. Gerdsen, P. Kroeger, “Digitalesignalverarbeitung in der Nachrichten-ubertragung”, Springer- Verlag,1993.[5] S. Haykin, “Communication Systems”,John Wiley & Sons, 1994.[6] L. R. Rabiner, B. Gold,”Theory andApplication of Digital Signal Processing”,Prentice-Hall, 1975[7]J. G. Proakis, D. G. Manolakis,“Introduction to Digital Signal Processing”,Macmillan Publishing Company, 1989[8] R. Volgers, “Phase Locked Loop Circuits,Philips Components”, 1989.[9] P. Hasan, “PLL FM DemodulatorPerformance Under Gaussian Modulation”,IEEE Trans. Commun., vol. COM-46,pp.437-440, April 1998.[10] P. Z. Peebles, “Probability, RandomVariables and Random Signal Principles”,McGraw -Hill, 1993.[11] TMS 320C40 User's Guide-TexasInstruments,1996.[12] D.Robertson, “Selelcting Mixed-SignalComponents for Digital Communicationssystems” Part V, Analog Dialogue 31-3(1997.)