APPENDIX B

GAUSSIAN PULSE-SHAPING FILTER

The Gaussian low-pass filter has a transfer function given by

H( f ) exp (a2f 2) (B:1)The parameter a is related to B, the 3-dB bandwidth of the baseband Gaussianshaping filter. It is commonly expressed in terms of a normalized 3-dB

bandwidth-symbol time product (BTs):

a

ln (2)p

2p Ts

BTs(B:2)

As a increases, the spectral occupancy of the Gaussian filter decreases and the impulseresponse spreads over adjacent symbols, leading to increased ISI at the receiver. The

impulse response of the Gaussian filter in the continuous-time domain is given by

h(t)

ppa

exp pa

t 2

(B:3)

which could easily be rearranged (Eq. B.4) to reveal its fit with the canonical form of a

zero-mean Gaussian random variable with standard deviation sh a

2p

p:

h(t) 12p

p(a

2p

p)exp t

2

2(a

2p

p)2

" #

(B:4)

Its integral from 1 to 1 is, of course, 1.

232

All-Digital Frequency Synthesizer in Deep-Submicron CMOS, by Robert Bogdan Staszewski andPoras T. BalsaraCopyright # 2006 John Wiley & Sons, Inc.

Let us now express the Gaussian filter in the discrete-time domain. Let t0 Ts=OSR be an integer oversample of the symbol duration and t kt0, k beingthe sample index. The discrete-time impulse response becomes

h(kt0)

ppa

exp pa

kt0

2

(B:5)

Substituting Eq. B.2 and dropping explicit dependence on t0 results in

hk

2pp

ln (2)p BTs

Ts|{z}

hmax

exp

2p

p

ln (2)p BTs k

OSR

2" #

(B:6)

The first factor in Eq. B.6 is the peak of the impulse frequency response:

hmax

ppa

2pp

ln (2)p BTs

Ts(B:7)

For BLUETOOTH, with BTs 0:5 and Ts 1ms, we obtain hmax 1:5054 MHz.For GSM, with BTs 0:3 and Ts 3:692 ms, we obtain hmax 244:62 kHz.

For reasons described in Chapter 5, it is more efficient to operate on the cumula-

tive coefficients

Ck X

k1

l0hl (B:8)

which could be precalculated and stored in a look-up table, with k 0 OSR 1 being the index. The minimum value of Ck is approximately zeroand the maximum value is approximately 1, since the integral of Eq. B.4 is

unity.

Figure B.1 shows the impulse hk, step Ck, and di-bit responses (differencebetween step and symbol-delayed step responses) of the BLUETOOTH GFSK

filter (BTs 0:5) with a length of three symbols, each symbol oversampledby 8.

Similarly, Fig. B.2 shows the impulse, step, and di-bit responses of the GSM

GMSK filter (BTs 0:3) with a length of four symbols, each symbol oversampledby 8. It reveals much more intersymbol interference (ISI) than in the case of

BLUETOOTH.

Figures B.3 and B.4 show the frequency responses of the BLUETOOTH and

GSM filters with varying filter lengths of three, four, and five symbols. A filter

length of three symbols is completely adequate for precise containment of the

APPENDIX B: GAUSSIAN PULSE-SHAPING FILTER 233

Figure B.1 Time response of a BLUETOOTH GFSK filter of four-symbol length

(BTs 0:3, OSR 8).

Figure B.2 Time response of a GSM GMSK filter of four-symbol length

(BTs 0:3, OSR 8).

234 APPENDIX B: GAUSSIAN PULSE-SHAPING FILTER

modulated output spectrum and sufficient attenuation of frequency components in

adjacent channels. However, due to the higher amount of ISI and much tougher

requirements for the modulated output spectrum, the GSM-standard filter would

require a filter length of at least four symbols.

Figure B.3 Frequency response of a BLUETOOTH GFSK filter for filter lengths of three,

four, and five symbols (BTs 0:5, OSR 8).

Figure B.4 Frequency response of a GSM GMSK filter for filter lengths of three, four, and

five symbols (BTs 0:3, OSR 8).

APPENDIX B: GAUSSIAN PULSE-SHAPING FILTER 235

Figure B.5 shows the spectrum of the baseband GMSK filter output FCW and RF

port R{e ju} with pseudorandom input data, in which

D f k FCWk fR2WF

(B:9)

and

u k 2pOSR

X

k1

l0D f k (B:10)

Figure B.5 Baseband (top) and RF (bottom) spectra of GMSK filter output with

pseudorandom input (five-symbol length, BTs 0:3, OSR 96).

236 APPENDIX B: GAUSSIAN PULSE-SHAPING FILTER