Unit 3 Spread Spectrum Modulation

Transcription

1 Unit 3 Spread Spectrum Modulation Instructor: Sau-Hsuan Wu Dept. of Electrical and Computer Eng., NCTU 1

2 What is spread spectrum modulation? A means of transmission in which data sequences occupy a bandwidth (BW) in excess of the minimum necessary BW Why do we need spread spectrum modulation? Interference rejection in multiple access channels Secure communications in a hostile environment where a transmitter may attempt to jam the transmission Dept. of Electrical and Computer Eng., NCTU 2

3 How do we do spread spectrum modulation? A straightforward method is to multiply the message symbol by a wideband pseudo noise (PN) spreading sequence Direct-Sequence spread spectrum (DSSS) modulation Available from blogspot.com Dept. of Electrical and Computer Eng., NCTU 3

4 Denote the transmit DSSS signal by m(t) = c(t)b(t) c(t) stands for the wideband PN signal b(t) is the narrowband message signal The received signal is r(t) = m(t) + i(t) = c(t)b(t)+ i(t) i(t) is an additive interference Dept. of Electrical and Computer Eng., NCTU 4

5 How do we do demodulation for DSSS? The same PN spreading code is used in the receiver to despread the received signal Available from blogspot.com Dept. of Electrical and Computer Eng., NCTU 5

6 Suppose that the receiver operates in perfect synchronism with the transmitter The multiplier output z(t) = c(t)r(t) = c 2 (t)b(t)+ c(t)i(t) Since c 2 (t) = 1, we have z(t) = b(t) + c(t)i(t) The spreading code c(t) will affect the interference i(t) just as it did the original signal b(t) Applying z(t) to a low-pass filter with a BW just large enough to accommodate b(t), most of c(t)i(t) is filtered out The low-pass filtering action is performed by the integrator Dept. of Electrical and Computer Eng., NCTU 6

7 DSSS with coherent BPSK Dept. of Electrical and Computer Eng., NCTU 7

8 In normal form, spectrum spreading is performed prior to phase modulation For the purpose of analysis, it is more convenient to exchange the order of PN code generator and BPSK modulator, leading to the following structure y(t) = x(t) + j(t) u(t) = c(t)y(t) = c 2 (t)s(t) + c(t)j(t) = s(t) + c(t)j(t) Dept. of Electrical and Computer Eng., NCTU 8

9 Signal-Space dimensionality and processing gain Consider the set of orthonormal basis functions: 2 cos 2, 1 0, otherwise 2 sin 2, 1, 0,1,2, 1 0, otherwise where T c is the chip duration, and N is the number of chips per bit The transmitted DS/BPSK signal x(t) may be expressed as 2 cos 2 where {c 0, c 1, c N-1 } denotes the PN sequence, with c k = 1 Dept. of Electrical and Computer Eng., NCTU 9

10 The transmitted signal x(t) is therefore N-dimensional in that it requires a minimum of N orthonormal basis functions for its representation For the jamming signal j(t), however, it has no knowledge of the signal phase of x(t) the spreading code of x(t), rather only has the bandwidth of x(t) According, we may represent the jammer j(t) by a general form where, 0,,1, 0,,1, 0 Thus, j(t) is 2N-dimensional, thought x(t) is N-dimensional Dept. of Electrical and Computer Eng., NCTU 10

11 The average power of j(t) is given by Due to the lack of knowledge of signal phase, the best strategy a jammer can apply is to place equal energy in and 2 Based on the above results, we next investigate the SNRs measured at the input and output of the DS/BPSK receiver Recall the despread received signal u(t) = s(t) + c(t)j(t) The coherent detector output is expressed as v 2 utcos2 Dept. of Electrical and Computer Eng., NCTU 11

12 The signal component in the detector output is given by 2 stcos2 2 cos 2 The plus sign corresponds to symbol 1, and the minus corresponds to symbol 0 Supposing that f c is an integer multiple of 1/T b The component from the jamming signal, on the other hand, is given by 2 c t tcos2 Where T c is the chip duration tϕ Dept. of Electrical and Computer Eng., NCTU 12

13 Now, approximate the PN sequence {c k } as an i.i.d. binary sequence, and the jammer to be fixed for a long period of time Define R.V. V cj and C k with sample values v cj and c k, respectively For a fixed jammer j k, we may define Given that P(C k = 1) = P(C k = 1) = ½, we have E[C k j k j k ] = P(C k = 1) j k P(C k = 1) j k = ½ [j k j k ] = 0 Consequently, for a fixed jammer: j =[j 0 j 1,, j N-1 ], we have Var V cj j 1 2 Where N = T b / T c is referred to as the spreading factor As a result, the output SNR of the detector follows SNR Var V cj j 2 Dept. of Electrical and Computer Eng., NCTU 13

14 On the other hand, the average signal power at the receiver input is, which gives the input SNR of the receiver as SNR Therefore, it follows that SNR 2 SNR This motivates us to define a processing gain (PG) of T b /T c as the gain in SNR obtained by the use of spread spectrum The longer is the PN sequence, (or the smaller the chip time T c is), the larger will be the PG We may also express the output SNR in decibel as 10log 10 (SNR O ) = 10log 10 (SNR I ) log10(PG) db The 3 db term results from the use of coherent detection Dept. of Electrical and Computer Eng., NCTU 14

15 Probability of error Recall the detector output v = v s + v cj, where Let v be the sample value of a RV V The signal model of the coherent detector of DS/BPSK is The BER is 0 0 is sent Therefore, P e depends on the PDF of By the central limit theorem, V cj is approximately Gaussian with zero mean and Var V cj j /2, i.e. V cj ~N(0, JT c /2) when N is large Making use of the BER of BPSK, similarly we have 1 2 erfc 1 2 erfc Dept. of Electrical and Computer Eng., NCTU 15

16 Antijam characteristic We may relate the BER of DS/BPSK to that of the typical BPSK by considering N 0 /2 = JT c /2 Given that the symbol energy is PT b = E b, we may also interpret the typical SNR as / Let (E b /N 0 ) min be the the minimum SNR required to support a prescribed BER We therefore may define a jamming margin, J/P, which is related to (E b /N 0 ) min by the following form 10log The higher the PG, the greater the ability to combat jamming Dept. of Electrical and Computer Eng., NCTU 16

17 Ex. 1 A spread spectrum system has the following parameters T b = ms and T c = 1us The processing gain (PG) = 4095 Assume that we need P e 10-5 For BPSK, when E b /N 0 = 10, 1 2 erfc 1 2 erfc log 10log log db Information bits can be detected reliably even when the noise or interference at receiver input is times the received signal power Dept. of Electrical and Computer Eng., NCTU 17

18 For DSSS, spreading is achieved instantaneously by using a PN sequence to modulate a PSK/QAM/FSK signal Anti-jam capability is determined by the processing gain The processing gain can be made larger with: A narrow chip duration or more numbers of chips per bit What if the spreading gain is still not large enough to overcome the effects of jammer? Force the jammer to cover a wider spectrum by randomly hopping the carrier from one frequency to the next Frequency-Hop (FH) Spread Spectrum The spectrum of the transmitted signal is spread sequentially Pseudo-random-ordered sequence of frequency hops Dept. of Electrical and Computer Eng., NCTU 18

19 A common modulation format for FH systems is M-ary FSK FH/MFSK Slow-Frequency Hopping: Several symbols are transmitted on each frequency hop, i.e. symbol rates R s = n hop rate R h, nn Fast-Frequency Hopping: R h = n R s, i.e, the carrier frequency will hop several times during the transmission of one symbol For a k-bit PN code, there are 2 k FHs For Fast-FH, we use noncoherent detection only because frequency synthesizers are unable to maintain phase coherent over multiple hops Dept. of Electrical and Computer Eng., NCTU 19

20 Ex. 2: Illustration of a slow FH/MFSK system # of bits/sym: K = 2 MFSK tones: M =2 K = 4 Len of PN seq. : k = 3 # of FH: 2 k = 8 For non-coherent MFSK B = 2 K R s W c = 2 k B B Dehopped frequencies Dept. of Electrical and Computer Eng., NCTU 20

21 For FH/MFSK systems, an individual tone of the shortest duration is referred to as a chip (not the DSSS chip interval) The chip rate is defined by R c = max (R h, R s ), where R h is the hop rate and R s is the symbol rate For a slow FH/MFSK signal, each symbol is a chip, such that R c = R s = R b /K R h, where K = log 2 M At each hop, the MFSK tones are still separated in frequency by an integer multiple of the symbol rate, also the chip rate As such, orthogonality is still maintained in a slow FH/MFSK The jamming signal has an effect on the FH/MFSK, in terms of SER, equivalent to that of AWGN on an MFSK system We, thus, may use the BER of MFSK for approximate evaluation of the SER in the FH/MFSK system Dept. of Electrical and Computer Eng., NCTU 21

22 Assuming that the jammer spreads its power J over the entire FH spectrum, the jammer s effect is equivalent to an AWGN with N 0 = J/W c, and W c is the FH bandwidth The FH/MFSK system is thus characterized by the symbol energy-to-noise spectral density ratio of / / The ratio of P/J is the reciprocal of the jamming margin For an MFSK with a frequency spacing, the output BW is 2 The processing gain (PG) of a FH/MFSK system is defined as W c /B = 2 R s 2 R s 2 PG 10 log 3 10log This PG assumes the jammer spreads its power over the entire FH spectrum However, if a jammer decides to concentrate on just a few of the hopped frequencies, the PG would be less than 3k decibels. Dept. of Electrical and Computer Eng., NCTU 22

23 For a fast FH/MFSK signal, each hop is a chip, such that there are multiple hops per M-ary symbol Noncoherent detection is used for data recovery For each FH/MFSK symbol, two procedures may be considered for detection A majority vote approach: Separate decisions are made on the K FH chips received A majority vote is used to make an estimate of the dehopped MFSK symbol A maximum likelihood approach: For each FH/MFSK symbol, likelihood functions are computed as functions of the total signals received over K chips The largest one is selected This is optimal in the sense that it minimizes the average SER Dept. of Electrical and Computer Eng., NCTU 23

24 Ex. 2: Illustration of a fast FH/MFSK system # of bits/sym: K = 2 MFSK tones: M =2 K = 4 Len of PN seq. : k = 3 # of FH: 2 k = 8 R c = R b = KR s Min tone spacing is KR s and B = 2 K KR s PG Dehopped frequencies 2 B Dept. of Electrical and Computer Eng., NCTU 24

25 HW5 (due on 6/9) 7.7, 7.10, 7.11, 7.12, and 7.13 Dept. of Electrical and Computer Eng., NCTU 25