Gaussian FIR pulse-shaping filter design
Specify that the modulation used to transmit the bits is a Gaussian minimum-shift keying (GMSK) pulse. This pulse has a 3-dB bandwidth equal to 0.3 of the bit rate. Truncate the filter to 4 symbols and represent each symbol with 8 samples.
bt = 0.3; span = 4; sps = 8; h = gaussdesign(bt,span,sps); fvtool(h,'impulse')
bt— 3-dB bandwidth-symbol time product
Product of the 3-dB one-sided bandwidth, in hertz, and the symbol
time, in seconds. Specify this value as a positive real scalar. Smaller
bt produce larger pulse widths.
span— Number of symbols
Number of symbols, specified as a positive integer scalar.
sps— Samples per symbol
Number of samples per symbol period (oversampling factor), specified as a positive integer scalar.
h— FIR filter coefficients
FIR coefficients of the Gaussian pulse-shaping filter, returned as a row vector. The coefficients are normalized so that the nominal passband gain is always 1.
The impulse response of the Gaussian filter is given by
BT is the bandwidth-symbol time product specified in
bt, where B is the 3-dB bandwidth of the
filter and T is the symbol time. The number of symbols between the
start and end of the impulse (
span) and the number of samples per
sps) determine the length of the impulse response:
For more information, see FIR Gaussian Pulse-Shaping Filter Design.
 Krishnapura, N., S. Pavan, C. Mathiazhagan, and B. Ramamurthi. “A baseband pulse shaping filter for Gaussian minimum shift keying.” Proceedings of the 1998 IEEE International Symposium on Circuits and Systems. Vol. 1, 1998, pp. 249–252.
 Rappaport, Theodore S. Wireless Communications: Principles and Practice. 2nd Ed. Upper Saddle River, NJ: Prentice Hall, 2002.