In this chapter, we apply the proposed algorithm to wireless applications. The bandpass signals considered in the simulations are GSM 900 (935-960 MHz, one-sided bandwidth 25 MHz), GSM 1800 (1805-1880 MHz, one-one-sided bandwidth 75 MHz) [17], DAB Eureka-147 L-Band (1472.286-1473.822 MHz, one-sided band-width 1536 KHz) [18], IEEE 802.11g (2412-2432 MHz, one-sided bandband-width 20 MHz) [19], and WCDMA (2119-2124 MHz, one-sided bandwidth 5 MHz).
6.1 Complexity Comparisons of The Proposed Algorithm and Previously Reported Meth-ods
In this section, we compare the number of addition and multiplication for finding the minimum sampling frequency using method in [14], method in [15], and our proposed method. Table 6.1 lists the complexity in finding the minimum sam-pling frequency for different combinations of wireless systems. The simulation result demonstrates that the proposed method can reduce the number of addi-tions and multiplicaaddi-tions significantly. The required numbers of addiaddi-tions and multiplications are reduced respectively by around 35-41% and 36-53% in two bandpass signals case, 28-58% and 61-76% in three bandpass signals case. As an-other example, we consider GSM 900 application with spectrum divided to 125 users and the bandwidth of each user is 200 kHz. Table 6.2 lists the complexity for
Case Method in [14] Method in [15] Proposed Method
ADD MUL ADD MUL ADD MUL
GSM900, GSM1800 29 50 38 50 17 28
DAB, 802.11g 65 122 161 296 42 78
GSM900, WCDMA 35 62 101 176 21 29
DAB, WCDMA 119 230 452 878 77 141
GSM900, GSM1800, 802.11g 105 186 87 109 60 42
DAB, GSM1800, 802.11g 75 126 99 133 41 36
GSM900, DAB, WCDMA 183 342 198 331 77 84
Table 6.1: Complexity for finding the minimum sampling frequency of multiple bandpass signals in terms of additions (ADD) and multiplications (MUL).
User Index Method in [14] Method in [15] Proposed Method
ADD MUL ADD MUL ADD MUL
‘57’, ‘101’ 329 650 3585 7144 11 13
‘37’, ‘49’ 1109 2210 3554 7082 15 22
‘10’, ‘45’, ‘88’ 2553 5082 4772 9479 1014 962
‘30’, ‘50’, ‘95’ 4359 8694 4780 9495 2325 2332
‘8’, ‘46’, ‘74’, ‘102’ 3270 6492 6001 11884 812 575
‘25’, ‘39’, ‘77’, ‘125’ 8952 17856 6013 11908 4525 3089
‘25’, ‘50’, ‘75’, ‘100’, ‘125’ 2336 4592 7253 14317 1734 1174
‘11’, ‘39’, ‘78’, ‘110’, ‘119’ 12062 24044 7251 14313 4164 2172
‘20’, ‘40’, ‘60’, ‘80’, ‘100’, ‘120’ 3765 7410 8488 16702 2476 1021
‘8’, ‘15’, ‘36’, ‘73’, ‘99’, ‘111’ 19593 39066 8467 16660 2986 1206 Table 6.2: Complexity for finding the minimum sampling frequency for GSM 900 with multiple users. For the ‘i’-th user, f`i = 935 + 0.2(i − 1) Mhz, Wi = 200 kHz, i = 1 − 125.
finding the minimum sampling frequency. The required numbers of additions and multiplications are reduced to around 25-98% and 73-99% compared to the other two methods. We can see that the proposed algorithm is much more efficient for finding the minimum bandpass sampling frequency.
Case GB (MHz) fs,min (MHz) ADD MUL
GSM900, GSM1800 0 240 17 28
12.5 240 15 13
DAB, WCDMA 0 13.9737 77 141
0.768 15.3357 50 88
GSM900, GSM1800, 802.11g 0 320 60 42
10 320 37 20
GSM900, DAB, WCDMA 0 77.2364 77 84
0.768 80.1509 70 71
Table 6.3: Complexity for finding the minimum sampling frequency with and without guard band.
6.2 Complexity Comparisons for Finding the Min-imum Sampling Frequency with and with-out Guard Band
Table 6.3 lists the complexity with and without guard band. The simulation results shows that introducing a larger guard band as larger, the minimum sam-pling frequency is in general larger. Furthermore, having guard band between different bandpass signals may increase or decrease the complexity, depending on the length of guard band. Consider the case of the signals that consists of three bandpass signals, GSM900, GSM1800, and IEEE 802.11g. We apply the proposed algorithm to find the minimum sampling frequency fs,min. It is equal to 320 MHz. Fig. 6.1 shows the folded spectrum. When GB = 10 MHz, the minimum sampling frequency is also equal to 320 MHz. In this particular exam-ple, including a guard band of 10 MHz does not increase the minimum sampling frequency. In fact we can see in Fig. 6.1 that, the replicas of different bandpass signals are spaced apart by at least 13 MHz. Applying the proposed algorithm with GB ≤ 13 MHz will yield the same fs,min.
0 320 f
GSM900, GSM1800, 802.11g 4864 320
GSM900, DAB, WCDMA, 802.11g 4864 137.1429
Table 6.4: Minimum sampling frequency comparisons with and without an or-dering constraint.
6.3 Minimum Sampling Frequency Comparisons with and without Ordering Constraint
Table 6.4 lists the minimum sampling frequency when there is a ordering among the replicas [13]. The constraint is such that in the [0, fs) frequency range the replica of Xi+(f ) is at the left of Xi+1+ (f ). We have also listed the minimum sampling frequency obtained using the proposed iterative algorithm without an ordering constraint. We can see that the minimum sampling frequency without an ordering constraint can be much smaller that with an ordering constraint.
Consider a signal that consists of two bandpass signals, GSM 900 and GSM 1800. The minimum sampling frequency is 240 MHz without an ordering con-straint as shown in Fig. 6.2(a), and 417.778 MHz with an ordering concon-straint as shown in Fig. 6.2(b). Since the proposed algorithm does not impose the ordering constraint, it can obtain the true minimum sampling frequency.
0 without an ordering constraint. (b) fs,min = 417.778 MHz with an ordering constraint.
Case Valid sampling frequency range (MHz) GSM900, GSM1800 240.0000 ≤ fs ≤ 240.6667
DAB, 802.11g 46.7880 ≤ fs ≤ 46.7986 GSM900, WCDMA 64.3636 ≤ fs ≤ 64.3889
DAB, WCDMA 13.9737 ≤ fs ≤ 13.9739
GSM900, GSM1800, 802.11g 320.0000 ≤ fs ≤ 321.6000 DAB, GSM1800, 802.11g 209.6139 ≤ fs ≤ 209.7391 GSM900, DAB, WCDMA 77.2364 ≤ fs ≤ 77.2667
Table 6.5: Valid ranges of the sampling frequency.
6.4 Range of Valid Sampling Frequency
Table 6.5 lists a valid sampling frequency range for different combinations of wire-less systems. Consider a signal that consists of GSM 900, GSM 1800, and IEEE 802.11g. The minimum sampling frequency is 320 MHz as shown in Fig. 6.3(a).
When we gradually increase the sampling frequency, the positive part of the replica of 802.11g moves towards fs/2. When we increase the sampling frequency to 321.6MHz, the positive part and negative part of replica of 802.11g will be aliasing if we keep increasing the sampling frequency as shown in Fig. 6.3(b). The frequencies in the range of 320-321.6 MHz are all alias-free sampling frequency.
0
Chapter 7 Conclusions
In this thesis, we have proposed an efficient algorithm for finding the minimum sampling frequency for signals that contain multi-passband signals. We have de-rived a set of necessary and sufficient conditions for alias-free sampling that can be checked with few computations. These conditions lead to an iterative algo-rithm for finding the minimum sampling frequency. This is done by iteratively increasing the sampling frequency to meet the alias-free conditions. The com-plexity for finding the minimum sampling frequency is much lower than existing methods. There is no need to consider ordering of the signal bands in the folded spectrum. The method can be easily extended the case when a guard band is required.
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