Bandwidth Considerations 1.5 2003/05/23 2003/09/08 11:51:06.935 GMT-5 Don Johnson dhj@rice.edu Don Johnson dhj@rice.edu Kyle Clarkson kclarks@rice.edu Elizabeth Gregory lizzardg@rice.edu Kevin Duh kevinduh@rice.edu Mariyah Poonawala mariyah@rice.edu Matthew Jeanes mjeanes@rice.edu Jeffrey Silverman jsilv@rice.edu stochasitc process random binary wave power density spectrum PSK ASK FSK bandwidth Kary Performance is not the only consideration in designing signal sets. The channel model has been assumed to have inifinite bandwidth: No matter what signal set is used by the trasnmitter, the waveforms of the signal portion of the received signal arrive at the receiver unaltered. Such a simple model may not be true and the bandwidth occupied by the signaling scheme is usually a consideration. The bandwidth occupied by an FSK scheme is difficult to analyze. Needless to say that the dominant contribution to this bandwidth is due to the frequency separation between the signals in the signal set. The bandwidth of ASK and PSK signal sets can be analyzed. The transmitted signal resulting from using these signal sets is given by the stochastic process s ASK t X t 1 2 2 E T 2 f 0 t θ s PSK t X t 2 E T 2 f 0 t θ where X t is a stochasitc process defined over each bit interval byTo obtain a stationary process, bit interval boundaries must undergo a (common) random shift τ uniformly distributed over 0 T . X t 1 12 -1 12 The values of X t in each bit interval are statistically independent. This process is termed the random binary wave. The power density spectrum of X t is given by S X f 1 T 2 f T 2 f 2 This power density spectrum corresponds to the baseband component of PSK signaling. The baseband power density spectrum corresponding to ASK signaling is given by S ASK f 1 4 δ f 1 T 2 f T 2 f 2 The power density spectra corresponding to the modulated signal sets consist of these power density spectra centered about the carrier frequencies ± f 0 . Consequently, the bandwidth occupied by these signaling schemes can be evaluated by considering their baseband counterparts. In general terms, the bandwidth occupied by both signal sets is infinite: These power density spectra are not bandlimited. In practical terms, the bandwidth is defined as the range of frequencies which contains a specified percentage of the total power. Using this definition, the ASK signal set occupies less bandwidth because of the presence of the DC component in the baseband power density spectrum. The signal set occupying the smallest bandwidth (ASK) has the worst performance (largest P e ). The antipodal signal sets occupy the largest bandwidth and have the best performance. Consequently, we encounter a typical engineering design tradeoff: Better performance can be obtained at the expense of bandwidth and vice versa.

K-ary Signal Sets To generalize these results to K-ary signal sets is obvious. The optimum receiver computes i i 0 … K 1 ϒ i r N 0 2 π i s i r s i 2 2 for each i and chooses the largest. Conceptually, these are no more complicated than binary signal sets. The minimum probability of error receiver remains a matched filter and has a similar structure to those shown previously. However, the computation of the probability of error may not be simple.

bandwidth The range of frequencies which contains a specific percentage of the total power.