Digital Communication Digital Modulation Schemes

Transcription

1 Digital Communication Digital Modulation Schemes Yabo Li Fall, 2013

2 Chapter Outline Representation of Digitally Modulated Signals Linear Modulation PAM PSK QAM Multi-Dimensional Signal Non-linear Modulation CPM Spectrum of the Digitally Modulated Signals 1

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4 Important Parameters in Digital Modulation Symbol Duration/Interval: The time T s in seconds that is used to transmit the signal waveform s m (t). Bit Duration/Interval: The time T b in seconds that is used to transmit one bit: Symbol Rate: T b = T s k. R s = 1 T s symbols/sec. Bit Rate: R = kr s = R s log 2 M bits/sec. 2

5 Important Parameters in Digital Modulation Average Signal Energy: E avg = M m=1 p m E m, E m is the energy of s m (t), p m is the probability of the m-th signal. p m = M 1 for equal probable signals. Average Bit Energy: E bavg = E avg k = E avg log 2 M, Average Power: P avg = E bavg T b = E avg T s. 3

6 Memoryless Modulation vs. Modulation with Memory Memoryless means the mapping from {a n } to {s m (t)} doesn t depend on previously transmitted waveform. Modulation with memory is vice versa. Linear Modulation vs. Non-linear Modulation Linear means the principal of superposition applies in the mapping from {a n } to {s m (t)}. Linear Non-Linear Memoryless PAM, PSK, QAM? Memory DPSK CPM, MSK, CPFSK 4

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8 PAM: The Signal Space Representation In signal space, the PAM modulated signal has only 1 dimension. s m (t) = s m f(t) f(t) = 2Eg g(t)cos2πf ct s m = A m E g 2 where E g = T 0 g 2 (t)dt The Euclidean distance between any pair of signal points is: d mn = s m s n 2 Eg = A m A n 2 The minimum distance between the signal points is: d min = 2E g 6

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10 PSK (Phase Shift Keying) The Transmit Waveform is: s m (t) = R [ e j2π(m 1) M = g(t)cos ] g(t)e j2πf ct 2π(m 1) M Signal Space Representation: cos2πf c t g(t)sin s m (t) = s mi f 1 (t) + s mq f 2 (t) f 1 (t) = 2Eg g(t)cos2πf ct 2π(m 1) M sin2πf c t f 2 (t) = 2Eg g(t)sin2πf ct s m [s mi s mq ] = Eg 2 cos 2π(m 1) M Eg 2 2π(m 1) sin M 8

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12 QAM (Quadrature Amplitude Modulation) The transmit waveform: s m (t) = R { (A mi + A mq )g(t)e j2πf ct } = A mi g(t)cos2πf c t A mq g(t)sin2πf c t Signal Space Representation: s m = [s mi s mq ] = A mi Eg 2 A mq Eg 2 The distance between two points d mn = s m s n = Eg 2 [ (Ami A ni ) 2 + (A mq A nq ) 2] 10

13 QAM (Quadrature Amplitude Modulation) The minimum distance: Assume A mi and A mq take the set of discrete values {(2m 1 M), m = 1,2,, M}, then d min = 2E g The average energy of the points E avg = 1 M E g 2 M m=1 (A 2 mi + A2 mq ) The d min, E avg and the kissing number determine the performance of the modulation scheme in the AWGN channel. 11

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15 Multi-dimensional Signal Modulation is the procedure that maps the information bits to the waveforms that to be transmitted to the channel. For PAM, PSK or QAM, the bits are mapped to a real or complex scalar value. For multi-dimensional signal, the bits can be mapped to a real or complex vector. Practical multi-dimensional modulations include time domain multi-dimensional signals, frequency domain multidimensional signals and code domain multi-dimensional signals, etc. 13

16 Multi-dimensional Signal Orthogonal Signaling The M transmitted signals are orthogonal, i.e., { E m = n < s m (t), s n (t) >= 1 m, n M 0 m n < s m (t), s n (t) > means the inner product between s m (t) and s n (t). The orthonormal basis in the signal space is: f i (t) = s j(t) E, 1 j N. Obviously, for orthogonal signaling, the dimension N is equal to the number of signals M, i.e., N = M In signal space, the signal vector can be represented as: s 1 = ( E,0,0,,0) s 2 = (0, E,0,,0). =. s M = (0,0,,0, E) 14

17 Multi-dimensional Signal Orthogonal Signaling The signal pairs have equal distance and the distance is equal to the minimal distance, i.e., d mn = d min = 2E Practical Orthogonal Signaling Schemes Frequency Domain Orthogonal Signaling: FSK (Frequency Shift Keying), OFDM (Orthogonal Frequency Division Multiplexing) Time Domain Orthogonal Signaling: TDM (Time Division Multiplexing) Code Domain Orthogonal Signaling: CDM (Code Division Multiplexing) 15

18 Multi-dimensional Signal Biorthogonal Signaling M biorthogonal signals are constructed from M 2 orthogonal signals by including the negative of the orthogonal signals. The dimension is equal to half of the number of signals, i.e., N = M 2. The correlation factor between any pair is: ρ s m s n s m s n = 0 or 1, m n. The minimum distance is: d min = 2E. 16

19 Multi-dimensional Signal Simplex Signaling Assume the M orthogonal signals in vector representation is s m, m = 1,2,, M, their mean is: s = 1 M The simplex signal is defined as: M s m m=1 s m = s m s, m = 1,2,, M The dimension of the simplex signal is: N = M 1 The energy per waveform is: s m 2 = s m s 2 = E ( 1 1 ) M The minimum distance is d min = 2E. 17

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22 Non-Linear Modulation with Memory: CPM The TX waveform is: s(t) = 2ε T cos(2πf ct + φ(t; I) + φ 0 ) φ 0 : Initial phase of the carrier φ(t; I): Time varying phase of the carrier φ(t; I) = 2π n k= I k h k q(t kt), nt t (n + 1)T I k {±1, ±3,, ±(M 1)} is the sequence of the information symbols. h k : modulation index, gives the step size of the change of the phase. q(t): normalized waveform shape, defines the shape of the change of the phase. 20

23 Modulation Index Define the step of the change of the phase between continuous symbols. h k = h, then the modulation index is fixed for all symbols Multi-h CPM: h k changes with k. Full Response and Partial Response q(t) define the shape of the change of the phase. q(t) = t 0 g(τ)dτ { g(t) = 0, for t > T, full response g(t) 0, for t > T, partial response 21

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25 CPFSK (Continuous Frequency Shift Keying) A special case of CPM: φ(t; I) = 2π n k= I k h k q(t kt), nt t (n + 1)T Modulation Index is: h k = h = 2f d T Phase Change Shape is: q(t) = 0 t 0 t 2π 0 t T 1 2 t T Substitute h and q(t) into φ(t; I), we can get: φ(t; I) = 2πf d T n 1 k= = θ n + 2πhI n q(t nt) I k + 2πf d q(t nt)i n 23

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27 MSK (Minimum Shift Keying) A special case of CPFSK at h = 1/2 When h = 1 2, f d = 1 4T, f = 2f d = 1 2T It is the minimum frequency separation that ensures the orthogonality of the two signals in a interval of T. GMSK (Gaussian Minimum Shift Keying) h = 1/2 with Gaussian pulse shape { [ ( g(t) = Q 2πB t T ) / ] ln2 Q 2 Q(t) is the Gaussian tail function. GMSK is used in the GSM system. [ 2πB ( t + T ) / ]} ln2 2 25

28 Spectrum Characteristics of the Digitally Modulated Signals The band-pass signal s(t) can be represented as s(t) = R { s l (t)e j2πf ct } s l (t): Equivalent low-pass signal. The auto-correlation of s(t) is: φ ss (τ) = R{φ sl s l (τ)e j2πf ct } The Fourier transform of φ ss (τ) can be written as: Φ ss (f) = 1 [ Φsl s 2 l (f f c ) + Φ sl s l ( f f c ) ] Φ sl s l (f) is the PSD of the low-pass equivalent signal s l (t). So in order to calculate Φ ss (f), it is sufficient to calculate Φ sl s l (f). 26

29 Spectrum for Linearly Modulated Signals For linear modulated signals, s l (t) can be written as: s l (t) = + n= I n g(t nt) I n : the sequence of symbols results from the mapping of k bits. I n is real for PAM and complex for PSK and QAM. g(t) the pulse shape. The auto-correlation of s l (t) is: φ sl s l (t + τ; t) = E [ s l (t)s l(t + τ) ] = + + n= m= E [ I n I m] g (t nt)g(t + τ mt) 27

30 Spectrum for Linearly Modulated Signals Assume {I n } is wide-sense stationary with mean µ i and auto-correlation φ ii (m) = E [ I n I n+m ] Then φ sl s l (t + τ; t) can be written as: φ sl s l (t + τ; t) = = + n= + m= φ ii (m n)g (t nt)g(t + τ mt) φ ii (m) + n= g (t nt)g(t + τ nt mt) + n= g (t nt)g(t + τ nt mt) is cyclostationary with period T, so is φ sl s l (t + τ; t). 28

31 Spectrum for Linearly Modulated Signals Average φ sl s l (t + τ; t) over a single period φ sl s l (τ) = 1 T = T/2 T/2 φ s l s l (t + τ; t)dt + m= φ ii (m) + n= 1 T T/2 nt T/2 nt g (t)g(t + τ mt)dt Define φ gg (τ) = + g (t)g(t + τ)dt φ sl s l (τ) can be written as: It a convolution! φ sl s l (τ) = 1 T + m= φ ii (m)φ gg (τ mt) 29

32 Spectrum for Linearly Modulated Signals F{φ gg (τ)} = G(f) 2, G(f) = F{g(t)} F{φ ii (m)} Φ ii (f) = + m= φ ii (m)e j2πfmt Φ sl s l (f) = 1 T G(f) 2 Φ ii (f) Consider the case that the information symbols are real and mutually uncorrelated, then φ ii (m) can be written as: φ ii (m) = { σ 2 i + µ 2 i (m = 0) µ 2 i (m 0) Then Φ ii (f) is: Φ i i(f) = σ 2 i +µ2 i + m= e j2πfmt = σ 2 i +µ2 i T + m= δ ( f m T ) 30

33 Spectrum for Linearly Modulated Signals The spectrum Φ sl s l (f) is: Φ sl s l (f) = σ2 i T G(f) 2 + µ2 i T 2 + m= G ( m T ) 2 δ ( f m T ) The first term is continuous spectrum controlled by the pulse g(t). The second term is discrete spectrum with separation 1/T If µ i = 0, then the second term vanishes. So, it is desired to have zero mean information sequence. 31