A Study on Performance Analysis for Error Probability in Swsk Systems

556 Tae-Il Jeong, et al.: A STUDY ON PERFORMANCE ANALYSIS FOR ERROR PROBABILITY IN SWSK SYSTEMS A Study on Performance Analysis for Error Probability in SWSK Systems Tae-Il Jeong, kwang-seok Moon, and Jong-Nam Kim, Member, KIMICS Abstract— This paper presents a new method for shift The simplest wavelet is Haar [4][5] and the keying using the combination of scaling function and wavelet communication signals named scaling wavelet shift keying (SWSK). An algorithm considered are amplitude shift keying (ASK), for SWSK modulation is carried out where the scaling frequency shift keying(FSK) and phases shift function and the wavelet are encoded to 1 and 0 in keying(PSK), which are transmitted over an AWGN accordance with the binary input, respectively. Signal energy, correlation coefficient and error probability of SWSK are channel. derived from error probability of frequency shift Amplitude shift keying, frequency shift keying and keying(FSK). The performance is analyzed in terms of error phase shift keying are the conventional digital probability and it is simulated in accordance with the kind of communication scheme. In frequency shift keying, the the wavelet. Based on the results, we can conclude that the frequency of the carrier varies in accordance with the proposed scheme is superior to the performance of the binary input. For binary transmission the carrier assumes conventional schemes. one frequency for 1 and another frequency for 0. This type of on-off modulation is called frequency shift keying. Index Terms— Correlation Coefficients, Frequency Shift Recently some attempts have been made to apply the Keying(FSK), Error Probability, Scaling Wavelet Shift Keying(SWSK), Wavelet Transforms. wavelet transform in digital communication [6][7]. Reference [8] provides the modulation scheme which uses the mother wavelet instead of the carrier frequency such as phase shift keying. That is, the phase of the mother I. INTRODUCTION wavelet is varied according to the source signal. The THE modulation schemes which use the scaling function and wavelet transform is mainly used in the field of wavelet, and the demodulation schemes which use the audio and image-signal processing. It is recently used in binary matched filter are introduced in [9][10]. digital communications. A promising application of In this paper, we propose a new shift keying using the wavelet transforms is in the field of digital wireless scaling function and the wavelet. The three parameters: communications where they can be used to generate signal energy, correlation coefficient and error probability waveforms that are suitable for transmission over wireless are also introduced, and are derived from the formulas of channels. This type of modulation is known as wavelet the frequency shift keying. The performance of SWSK is transform. The advantage of this scheme emerges from its analyzed in terms of the error probability, which is diversity strategy; wavelet modulation allows obtained from the average energy and correlation transmission of the data signal at multiple rates coefficient. simultaneously [1]. Wornell and Oppenheim outlined the This paper is organized as follows: Section Ⅱ reviews design of the transmitter and receiver for wavelet the mathematical preliminaries. An algorithm for modulation [2]. The performance of wavelet modulation modulator and demodulator for SWSK is presented in in an additive white Gaussian noise (AWGN) channel was Ⅲ also evaluated in Wornell’s work. Ptasinski and Fellman Section . Simulation results and performance analysis Ⅳ. simulated wavelet modulation using the Daubechies are presented in Section Finally, conclusions are wavelet and measured the performance of the bit error drawn in Section Ⅴ. rate (BER) using an AWGN channel. Ptasinski’s work showed the performance of wavelet modulation to be equivalent to that calculated by Wornell[3]. II. MATHEMATICAL PRELIMINARIES __________ Manuscript received July 11, 2011; revised August 1, 2011; accepted A. Error probability of frequency shift keying August 11, 2011. In frequency modulation, the frequency of the carrier Tae-Il Jeong and Kwang-Seok Moon are with the Department of Electronics varies in accordance with the binary input. For binary Engineering, Pukyong National University, Pusan, 608-737, Korea (Email: transmission the carrier assumes one frequency to be 1 tijeong@daum. net and [email protected] ) Jong-Nam Kim(Corresponding Author) is with the Department of IT and another frequency to be 0. This type of on-off Convergence & Application Engineering, Pukyong National University, Pusan, modulation is called frequency shift keying. The 608-737, Korea (Email:[email protected]) frequency shift keying signals are defined by [11][12] INTERNATIONAL JOURNAL OF KIMICS, VOL. 9, NO. 5, OCTOBER 2011 557 s (t) = Acosω (t), 0 ≤ t ≤ T , for 0 ⎧ 0 0 b Ψ(t) = 2∑ g(k) ϕ(2t − k) S FSK (t) = ⎨ (1) k ⎩ s1(t) = Acosω1(t), 0 ≤ t ≤ Tb , for 1 ϕ(t) = 2∑h(k) ϕ(2t − k ) where, s0(t) and s1(t) correspond to the binary symbols 0 k and 1, respectively. The average energy(E) and the ρ correlation coefficient( ) per bit for coherent matched This methodology for implementing the discrete filter is given as, respectively wavelet transform is known as multiresolution analysis. The filter coefficients have to satisfy the conditions as T 1 b follows[4][5]: E = []s 2 (t) + s 2 (t) dt (2) 2 ∫ 0 1 0 ∑h[n] = 2 (5a) T b n []s (t)s (t) dt 2 ∫ 0 1 ∑ h [n] =1 (5b) ρ = 0 (3) n E ∑ g[n] = 0 (6a) n The probability of error yields ∑ g 2[n] =1 (6b) n 1 ⎛ E(1− ρ) ⎞ 1 ⎛ A2T ⎞ Pe = erfc⎜ ⎟ = erfc⎜ b ⎟ (4) ⎜ ⎟ ⎜ ⎟ where h[n] is the scaling function and corresponds to 2 ⎝ 2η ⎠ 2 ⎝ 4η ⎠ the coefficient of the low pass filter, g[n] is the wavelet where, erfc is complementary error function, and and corresponds to the coefficient of the high pass filter. denotes the noise power in watt/Hz. A is the amplitude of We will now discuss a mothod for construction of carrier signals. It is assumed that the noise is additive scaling function using successive approximation. white Gaussian noise with a two-sided power spectral 2 i-th iteration density of η/2, mean zero and variance σ =η / 2 . For 2h0[n] 2 2h1[n] 2 2h2[n] 2h(i)[n] the coherent detection, matched filter detection is Fig. 1. Discrete impulse response for scaling functions. optimum as illustrated in [11][12]. B. Discrete Wavelet Transforms The discrete impulse response i The discrete wavelet transform algorithms make use of (i) h [n] = ∏*hk [n] (7) bank of quadrature filters. The filter coefficients are based k =0 on the mother wavelet, ψ(t), and the scaling function, φ(t) can be obtained in the i-th iteration step by i-fold derived from the mother wavelet. The mother wavelet and convolution of the dyadic upsampled inpulse response scaling function are used to implement high pass filters and low pass filters respectively. The procedure for ⎧ 2h[m] if n = 2k m decomposition of a signal using discrete wavelet hk [n] = ⎨ transform at first, involves the convolution of the signal ⎩0 otherwise with a pair of quadrature filters to obtain the The impulse response h(i)[n] has (N −1)(2i+1 −1) +1 approximation coefficient from the low pass filter and to obtain the detailed coefficients from the the high pass coefficients, where N is the number of the coefficient of filter. The outputed data sequences are then decimated by h[n]. If the number of iteration is incresed i = ∞ then a factor of two. To compute the next level of resolution, the discrete impulse response is considerd to be a the decimated sequence of detailed coefficients are continuous signal. Hence equ.(7) can be written as[4][5] inputted to the next set of quadrature filters. In this th ∞ (∞) fashion the wavelet coefficients at the n level of h [n] = ∏*hk [n] ≅ ϕ(t) (8) decomposion can be obtained[4][5]. k =0 Let the signal processing functions of the high pass Similarly filters and the low pass filters be denoted by the operators ∞ (∞) H and G. Now let {h(k)} be a square-summable sequence g [n] = ∏* gk [n] ≅ Ψ(t) (9) of coefficients which defines the linear operator H, and k =0 similarly {g(k)} for G. The relationship between the mother wavelet, the scaling function and the filter where, * means a convolution. coefficients are defined as [7] In practical applications, few iteration steps are enough to obtain the scaling function in Fig.1. 558 Tae-Il Jeong, et al.: A STUDY ON PERFORMANCE ANALYSIS FOR ERROR PROBABILITY IN SWSK SYSTEMS C. Cross-correlation Receiver represented in Fig. 3 which is similar to frequency shift The optimum receiver for a known signal in an keying. If the binary input is ‘1’ then the modulator is addititive white gaussian noise is the correlator or encoded to scaling function, and if the binary input is ‘0’ matched filter. The correlator performs a cross-correlation then encoded to the wavelet. of the received signal r(t), with each of the prototype of the transmitted signal sm(t), on the interval 0 ≤ t ≤Tb, 0 producing m outputs which are then compared by the ψ (t) detector. The decision circuit determines the largest T S (t) φ(t) b SWSK magnitude output from the output of the sampler and 1 declares it as the transmitted symbol [4]. Fig. 3. SWSK system modulator. Ⅲ. SCALING WAVELET SHIFT KEYING B. SWSK demodulation In SWSK demodulation, we find the sum of the A. SWSK modulation correlation for the interval 0 ≤ t ≤Tb, later we compare it The conventional modulation schemes for frequency with the given threshold. If the sum is larger than the shift keying require two carrier frequencies.

Load more