GAUSSIAN PULSE-SHAPING FILTER

Transcription

1 APPENDIX B GAUSSIAN PULSE-SHAPING FILTER The Gaussian low-ass filter has a transfer function given by H( f ) ¼ ex ( a 2 f 2 ) (B:1) The arameter a is related to B, the 3-dB bandwidth of the baseband Gaussian shaing filter. It is commonly exressed in terms of a normalized 3-dB bandwidth-symbol time roduct (BT s ): ffiffiffiffiffiffiffi ln (2) T a ¼ ffiffi s (B:2) 2 BT s As a increases, the sectral occuancy of the Gaussian filter decreases and the imulse resonse sreads over adjacent symbols, leading to increased ISI at the receiver. The imulse resonse of the Gaussian filter in the continuous-time domain is given by h(t) ¼ a ex 2 a t (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 s h ¼ a ffiffi 2 : " # 1 t 2 h(t) ¼ ffiffiffiffiffiffi ffiffiffi ex 2 (a 2 ) 2(a ffiffiffi (B:4) 2 ) 2 Its integral from 1 to 1 is, of course, 1. All-Digital Frequency Synthesizer in Dee-Submicron CMOS, by Robert Bogdan Staszewski and Poras T. Balsara Coyright # 2006 John Wiley & Sons, Inc. 232

2 APPENDIX B: GAUSSIAN PULSE-SHAPING FILTER 233 Let us now exress the Gaussian filter in the discrete-time domain. Let t 0 ¼ T s =OSR be an integer oversamle of the symbol duration and t ¼ kt 0, k being the samle index. The discrete-time imulse resonse becomes h(kt 0 ) ¼ a ex 2 a kt 0 (B:5) Substituting Eq. B.2 and droing exlicit deendence on t 0 results in ffiffiffi 2 h½kš ¼ffiffiffiffiffiffiffi ln (2) BT s T s fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} h max " ffiffi 2 # 2 k ex ffiffiffiffiffiffiffi BT s ln (2) OSR (B:6) The first factor in Eq. B.6 is the eak of the imulse frequency resonse: ffiffiffi h max ¼ a ¼ 2 BT s ffiffiffiffiffiffiffi ln (2) T s (B:7) For BLUETOOTH, with BT s ¼ 0:5 and T s ¼ 1 ms, we obtain h max ¼ 1:5054 MHz. For GSM, with BT s ¼ 0:3 and T s ¼ 3:692 ms, we obtain h max ¼ 244:62 khz. For reasons described in Chater 5, it is more efficient to oerate on the cumulative coefficients C½kŠ ¼ Xk 1 h½lš l¼0 (B:8) which could be recalculated and stored in a look-u table, with k ¼ 0 OSR 1 being the index. The minimum value of C½kŠ is aroximately zero and the maximum value is aroximately 1, since the integral of Eq. B.4 is unity. Figure B.1 shows the imulse h½kš, ste C½kŠ, and di-bit resonses (difference between ste and symbol-delayed ste resonses) of the BLUETOOTH GFSK filter (BT s ¼ 0:5) with a length of three symbols, each symbol oversamled by 8. Similarly, Fig. B.2 shows the imulse, ste, and di-bit resonses of the GSM GMSK filter (BT s ¼ 0:3) with a length of four symbols, each symbol oversamled by 8. It reveals much more intersymbol interference (ISI) than in the case of BLUETOOTH. Figures B.3 and B.4 show the frequency resonses of the BLUETOOTH and GSM filters with varying filter lengths of three, four, and five symbols. A filter length of three symbols is comletely adequate for recise containment of the

3 234 APPENDIX B: GAUSSIAN PULSE-SHAPING FILTER Figure B.1 Time resonse of a BLUETOOTH GFSK filter of four-symbol length (BT s ¼ 0:3, OSR ¼ 8). Figure B.2 Time resonse of a GSM GMSK filter of four-symbol length (BT s ¼ 0:3, OSR ¼ 8).

4 APPENDIX B: GAUSSIAN PULSE-SHAPING FILTER 235 Figure B.3 Frequency resonse of a BLUETOOTH GFSK filter for filter lengths of three, four, and five symbols (BT s ¼ 0:5, OSR ¼ 8). modulated outut sectrum and sufficient attenuation of frequency comonents in adjacent channels. However, due to the higher amount of ISI and much tougher requirements for the modulated outut sectrum, the GSM-standard filter would require a filter length of at least four symbols. Figure B.4 Frequency resonse of a GSM GMSK filter for filter lengths of three, four, and five symbols (BT s ¼ 0:3, OSR ¼ 8).

5 236 APPENDIX B: GAUSSIAN PULSE-SHAPING FILTER Figure B.5 Baseband (to) and RF (bottom) sectra of GMSK filter outut with seudorandom inut (five-symbol length, BT s ¼ 0:3, OSR ¼ 96). Figure B.5 shows the sectrum of the baseband GMSK filter outut FCW and RF ort R{e ju } with seudorandom inut data, in which and D f ½kŠ ¼FCW½kŠ f R 2 W F u ½kŠ ¼ 2 X k 1 D f ½kŠ OSR l¼0 (B:9) (B:10)