Analysis of Pulse Transmission System

Yu Lan Ding GuoningInstitute of Electronic Engineering

Radar AcademyWuhan 430019, Hubei, China

Abstract—This paper describes the jitter of output pulse frontedge with input noise, modulation and demodulation noise, phasenoise of local oscillators and channel noise for AM and FM pulsetransmission systems. The analytical results show that the totaljitter of system output pulse front edge includes three parts:input jitter, IF jitter and RF jitter, which can be regarded as thefoundation of design for IF and RF components of pulsetransmission systems.

tS

TKeywords- Jitter of pulse front edge; Input jitter; IF jitter; RF

jitter.

I. INTRODUCTION

The pulse transmission systems are widely adopted in radarand communication systems, in which the jitter of pulse frontedge is the key factor. Reference [5] ~ [16] illustrate jitter andphase noise, analyze the jitter and phase noise in mixers,oscillators, samplers, and PLL, discuss the relation between thejitter and BER of digital communication system. Reference [1]describes the relation between jitter of pulse front edge andcarrier-to-noise ratio for noiseless input signal and ideal analogpulse transmission systems. This paper further analyses the practical pulse transmission systems, deduces the formulae which describe the jitter of pulse front edge with input noise,modulation and demodulation noise, phase noise of localoscillators and channel noise for AM and FM pulsetransmission systems. The total jitter of pulse transmissionincludes three parts: input jitter, IF jitter and RF jitter, which may be regarded as the foundation of design for IF and RFcomponents of pulse transmission system. Section gives thestructure of AM pulse transmission system and analyzes itstransmission performance, section gives the structure of FMpulse transmission system and analyzes its transmissionperformance, and section concludes the analytical resultsand makes suggestions to the design of pulse transmissionsystems.

II. ANALYSIS OF AM PULSE TRANSMISSION SYSTEM

When a pulse string signal passes through a band pass transmission system, its front edges will incline to the right (as shown in Fig. 1). If the noise exists, the jitter occurs on therising-edge of pulse. In figure 1, denotes the pulse risingtime (from 10% to 90%), denotes the pulse amplitude,mA

denotes the pulse width, and T denotes the pulse period.

According to reference [1], the relation between meansquare value of jitter of pulse rising edge and signal-to-

noise ratio

2rmsJ

SN

can be described by

12

2

/ 0.8

rms

SN J

(1)

tSt tSortSrtSot

tM

tSm tSd

tV

PA

Pulse stringsignal

LNA

AmplitudeDemodulator

OutputSignal

Mixer

Fig.2, The composition block diagramof AM pulse transmission system

AmplitudeModulator

OSC MixerOSC

8.00

mA

t

……

Fig.1, waveform of a pulse stringsignal with linear rising front edge

978-1-4244-2251-7/08/$25.00 ©2008 IEEE

Figure 2 shows the practical structure of AM pulse transmission system. On the basis of analysis for its structure,we discuss the relation between the jitter of pulse front edgeand various noises in the transmission system such as inputnoise, modulation and demodulation noise, phase noise oflocal oscillators, and channel noise etc.

As shown in Fig.2, the input signal of amplitudemodulator may be expressed as tNtStM b , where

is the pulse string signal,tS tNb is the input noise.The output of amplitude modulator may be described as

, where

is the amplitude noise of the modulator,

cosm m cS t M t N t t tm

tNm tm is phase noise of the modulator.

The output of local oscillator in transmitter can be

expressed as , wherecosot o otS t t t tot is

the phase noise of the oscillator.The output of mixer in transmitter can be written as

cost m c o mS t M t N t t t tot

The received signal can be considered as the sum of transmitted signal and white Gaussian noisetSt tN , thatis

tNtStS tr

cosm c o m otM t N t t t t N tThe output of local oscillator in receiver can be expressed

as , where is the

phase noise of the oscillator.

cosor o orS t t t or t

The output of mixer in receiver can be described as

cosd m c m ot orS t M t N t t t t t n t

Where is narrowband Gaussian noise and can bedescribed by

tn

ttnttntn cscc sincosIf the total phase noise is small enough, that is

2m ot ort t t , we have

cos cos sin

cos sind m c c c s c

m c c s c

S t M t N t t n t t n t t

M t N t n t t n t tHence, the amplitude demodulated signal may be

described as 2 2

m c s dV t M t N t n t n t N t

Where denotes the noise of amplitude demodulator.dN tIf the carrier-to-noise ratio is large enough, that is

M t n t , we get

m c dV t M t N t n t N t

b m d cV t S t N t N t N t n t

Where bN t results from the input noise,

m dN t N t results from the noise of IF modulator and

demodulator, and cn t results from the channel noise of RF

receiver.The total output noise can be expressed mathematically

as

b m d cN t N t N t N t n tThe total output noise power can be calculated below

2b m d cN E N t N t N t n t

Since the noise bN t , , , and mN t dN t cn t are

uncorrelated with each other, thus2 2 2 2b m d cN E N t E N t E N t E n t

The power of the input noise, the IF modulation anddemodulation noise, and the RF channel noise can becalculated respectively as follows

The power of input noise:

2B

B b bB

N E N t f df (2)

The power of IF noise:2 2

IF m dN E N t E N tB B

m dB B

f df f df (3)

The power of RF noise:

202

B

RF c cB

N E n t f df Bn (4)

Where b f denotes the PSD of input noise,

m f denotes the PSD of IF modulation noise, d f

denotes the PSD of IF demodulation noise, c f denotes

the PSD of RF channel noise, and B denotes the base bandwidth of pulse transmission system.

The power of pulse signal can be calculated by2mS A (5)

Where stands for amplitude of pulse.mAWe may calculate the jitter of pulse rising edge by

computing the output signal-to-noise ratio firstly, that is

B IF R

S SN N N N F

(6)

According to equation (1), it can be simplified as 2 2 2

/ 0.8 / 0.8 / 0.8 / 0.8AM B IF RF

AM B IF RF

J J J J2

Where the B and are respectively the rising timeand jitter of a pulse string with additive noise which pass through a noiseless transmission system, the

BJ

IF and arerespectively the rising time and jitter of a noiseless pulsestring which pass through a practical IF self-loop transmissionsystem, and the

IFJ

RF and are respectively the rising timeand jitter of a noiseless pulse string which pass through a noiseless IF and practical RF transmission system.

RFJ

Generally AM B IF RF , above equation can besimplified as

2 2 2 2AM B IF RJ J J J F (7)

III. ANALYSIS OF FM PULSE TRANSMISSION SYSTEM

In similar manner like section II, we will discuss therelation between the jitter of pulse front edge and variousnoises in FM pulse transmission system.

As shown in Fig.3 the input signal of frequencymodulator may be expressed as tNtStM , where

is the pulse string signal,b

tS tNb is the input noise.The output of frequency modulator may be described as

cosm c c mS t A t t t , where tm is the

phase noise of the modulator, tt

FK M d , in which

FK M is pulse frequency modulation signal.

The output of local oscillator in transmitter can be

expressed as cosot o ott tS t , where tot is

the phase noise of the oscillator.The output of mixer in transmitter can be described as

cost c c o m otS t A t t t tThe received signal can be considered as the sum of

transmitted signal tS and white Gaussian noiset tN , that

is: tNtStS tr

cosc c o m otA t t t t N tThe output of local oscillator in receiver can be expressed

as cosor o orS t t t , where or t is the

phase noise of the oscillator.The output of mixer in receiver can be written as

cosd c c m ot orS t A t t t t t n t

Where tn is narrowband Gaussian noise and can bedescribed by

ttnttntn cscc sincosIf the carrier-to-noise ratio is large enough, that

is tnAc , we have

cos

cos sin

cos

d c c m ot or

c c s c

sc c m ot or

c

S t A t t t t t

n t t n t t

n tA t t t t t

ASince the output voltage of frequency demodulator is

directly proportional to the instantaneous frequency deviationof its input signal, that is

12

1 1 12 2 2

1 1 1 12 2 2

sm ot or d

c

mF F b d

ot or s

c

n tdV t t t t t N tdt A

d tK S t K N t N t

dtd t d t dn tdt dt A dt

In which the output signal can be described by

OutputSignal

tV

tSd

tSrtSor

tM

tSm

tSottSt

Pulse stringsignal

LNA

FrequencyDemodulator

Mixer

Fig.3, The composition block diagramof FM pulse transmission system

FrequencyModulator

OSC MixerOSC

PA

12o FS t K S t ,

max

12o F mS t K A f

So the output power of signal can be described by2

2

2F mK AS f (8)

The total output noise can be described by1 1

2 21 1 1

2 2 2

mF b d

ot or s

c

d tN t K N t N t

dtd t d t dn tdt dt A dt

So the total output noise power can be calculated as follows.

2

2

1 12 2

1 1 12 2 2

mF b d

ot or s

c

d tN E N t E K N t N t

dt

d t d t dn tdt dt A dt

Since the noise ,bN t md tdt

, ,dN t otd tdt

,

ord tdt

, and sdn tdt

are uncorrelated with each other,

thus22

2

2 2

1 12 2

1 1 12 2 2

mF b d

ot or s

c

d tN E K N t E E N t

dt

d t d t dn tE E E

dt dt A dt

2

Three types of noise power as section II can be calculatedby

2212 2

BF

B F b bB

KN E K N t f df (9)

22

2

12

mIF d

B B

m dB B

d tN E E N t

dt

f f df f df

(10)

2

2

1 12 2

12

ot orRF

s

c

d t d tN E E

dt dt

dn tE

A dt

2

32 2 0

2

23

B B

ot orcB B

n Bf f df f f dfA

(11)

In the similar way like section II, we have

B IF R

S SN N N N F

2

(12)

It can be simplified as 2 2 2FM B IF RFJ J J J (13)

IV. CONCLUSIONS

Equation (7) and (13) show that whether the pulsetransmission system adopts amplitude or frequency modulationmode, the output pulse jitter of the system includes three parts:the jitter due to input noise; the jitter due to IF modulation and demodulation noise; and the jitter duo to RF phase noise oflocal oscillators and channel noise. We may design and controlthe system parameters by above equations to meet the practicalrequirements; the design foundation of the pulse RFtransmission system may be founded in reference [1].

REFERENCES

[1] Yu Lan, “A Comparison of pulse Transmission Systems”, Proceedingsof ICCT 2006.

[2] Yu Lan, “A Comparison of Reference Source Signal TransmissionSystems”, Proceedings of ICCT 2006.

[3] Fan C X. “Theory of Communications”. China: National Defense and Industry Publishing House.

[4] Lu D J. “Stochastic Processes and Its Applications”. China: TsinghuaUniversity Publishing House.

[5] Maxim APPLICATION NOTE 3359, “Clock (CLK) Jitter and PhaseNoise Conversion” ,Sep 23, 2004.

[6] Maxim APPLICATION NOTE 3631, “Random Noise Contribution toTiming Jitter—Theory and Practice”, Sep 02, 2005.

[7] Maxim Application Note HFAN-4.0.2 (Rev. 1, 02/03), “Convertingbetween RMS and Peak-to-Peak Jitter at a Specified BER”, 07/28/03.

[8] Thomas H. Lee, Member IEEE, and Ali Hajimiri, Member IEEE,“Oscillator Phase Noise: A Tutorial”, IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 35, NO. 3, MARCH 2000.

[9] Joel Phillips and Ken Kundert, “Noise in Mixers, Oscillators, Samplers,and Logic: An Introduction to Cyclostationary Noise”, The Designer’sGuide Community Version 2, 9 September 2001.

[10] Ali Hajimiri, Sotirios Limotyrakis, and Thomas H. Lee, Member IEEE,“Jitter and Phase Noise in Ring Oscillators”, IEEE JOURNAL OFSOLID-STATE CIRCUITS, VOL. 34, NO. 6, JUNE 1999.

[11] Ken Kundert, “Modeling Jitter in PLL-based Frequency Synthesizers”,Designer’s Guide Consulting, Inc. Version 4g, August 2006.

[12] Ken Kundert, “Predicting the Phase Noise of PLL-Based FrequencySynthesizers”, Version 4e, August 2006.

[13] Ken Kundert, “Predicting the Phase Noise and Jitter of PLL-Based Frequency Synthesizers”, Version 4g, August 2006.

[14] Maxim APPLICATION NOTE 1916, “An Introduction to Jitter in Communications Systems”, Mar 06, 2003.

[15] Maxim Application Note HFAN-4.0.3 (Rev. 0, 09/01), “Jitter in Digital Communication Systems, Part 1”.

[16] Maxim Application Note HFAN-4.0.4 (Rev. 0, 03/02), “Jitter in Digital Communication Systems, Part 2”.