Qpsk maps a pair of data bits into a waveform

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Unformatted text preview: 2012 104 / 130 Lecture Notes 5 BPSK Modulation: Revisited BPSK An alternative view of BPSK is that of two antipodal signals; that is √ s0 (t ) = E ψ (t ), 0 ≤ t ≤ T and √ s1 (t ) = − E ψ (t ), 0≤t ≤T where ψ (t ) = 2/T cos(2π fc t ), 0 ≤ t ≤ T is a unit energy waveform. The above describes the signals transmitted only during the interval [0, T ]. Obviously this is repeated for other intervals. The receiver correlates with ψ (t ) over the interval [0, T ] and compares with a threshold (usually 0) to make a decision. The correlation receiver is shown below. EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 105 / 130 Lecture Notes 5 BPSK Modulation: Revisited BPSK r (t ) '$ Ed ET d 0 &% T E >γ <γ dec s0 dec s1 ψ (t ) This is called the correlation receiver. Note that synchronization to the symbol timing and oscillator phase are required. EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 106 / 130 Lecture Notes 5 Quaternary Phase Shift Keying (QPSK) Quarternary Phase Shift Keying (QPSK) Modulation BPSK maps a single data bit into a waveform (sinusoid) with one of two phases (0 or π ). QPSK maps a pair of data bits into a waveform (sinusoid) with one of four phases. QPSK doubles the data rate with the same bandwidth. EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 107 / 130 Lecture Notes 5 QPSK Modulator In-phase Data Source √ Quaternary Phase Shift Keying (QPSK) 2P cos(2π fc t ) bI (t ) s(t ) Quadrature-Phase Data Source bQ (t ) √ − 2P sin(2π fc t ) √ √ s(t ) = bI (t ) 2P cos(2π fc t ) − bQ (t ) 2P sin(2π fc t ) √ = = 2P cos(2π fc t + φ(t )) EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 108 / 130 Lecture Notes 5 Quaternary Phase Shift Keying (QPSK) 1 1 0.5 bI(t) 0.5 bQ(t) 0 −0.5 −1 0 0 −0.5 2 −1 0 4 time (s) 2 0.5 φ(t)/π 0 −1 −2 0 0 −0.5 2 4 time (s) EECS 455 (Univ. of Michigan) 4 1 1 s(t) 2 time (s) −1 0 2 4 time (s) Fall 2012 September 6, 2012 109 / 130 Lecture Notes 5 Quaternary Phase Shift Keying (QPSK) QPSK Demodulator 2/T cos(2π fc t ) In-Phase Branch Low Pass Filter I > 0 Data =+1 < 0 Data =-1 Q I Quadrature-Phase Branch Low Pass Filter − Q > 0 Data =+1 < 0 Data =-1 2/T sin(2π fc t ) EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 110 / 130 Lecture Notes 5 Quaternary Phase Shift Keying (QPSK) QPSK Q (-1,1) (1,1) I (-1,-1) EECS 455 (Univ. of Michigan) (1,-1) Fall 2012 September 6, 2012 111 / 130 Lecture Notes 5 Quaternary Phase Shift Keying (QPSK) QPSK, Eb/N0=40dB QPSK, Eb/N0=30dB 2 y (T) 0 s ys(T) 2 −2 0 −2 −2 0 y (T) 2 −2 c 2 c QPSK, Eb/N0=20dB QPSK, Eb/N0=10dB 2 y (T) 2 0 s ys(T) 0 y (T) −2 0 −2 −2 EECS 455 (Univ. of Michigan) 0 yc(T) 2 −2 Fall 2012 0 yc(T) 2 September 6, 2012 112 / 130 Lecture Notes 5 Quaternary Phase Shift Keying (QPSK) QPSK, Eb/N0=8dB QPSK, Eb/N0=6dB 2 ys(T) ys(T) 2 0 −2 0 −2 −2 0 y (T) 2 −2 c 2 c QPSK, Eb/N0=4dB QPSK, Eb/N0=2dB 2 ys(T) 2 ys(T) 0 y (T) 0 −2 0 −2 −2 EECS 455 (Univ. of Michigan) 0 yc(T) 2 −2 Fall 2012 0 yc(T) 2 September 6, 2012 113 / 130 Lecture Notes 5 Quaternary Phase Shift Keying (QPSK) QPSK Summary Good error probability performance: Pe,b = 10−5 at Eb /N0 = 9.6dB (good energy efficiency). Bandwidth decreases away from carrier only as 1/(f − fc )2 (poor bandwidth efficiency). Double the bandwidth efficiency of BPSK. Constant envelope signal (good for efficient use of power amplifiers). Low Complexity. Higher complexity than BPSK since two branches in modulator and demodulator are required EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 114 / 130 Lecture Notes 5 Spectrum of Passband Signals Spectrum for Passband Pulses Now let Z (t ) = Y (t ) cos(2π fc t + θ ) with θ uniform on [0,2π ] and independent of Y (t ). Then RZ (τ ) = SZ (f ) = 1 RY (τ ) cos(2π fc τ ) 2 1 [SY (f − fc ) + SY (f + fc )] 4 If Z (t ) = Y1 (t ) cos(2π fc t + θ1 ) + Y2 (t ) cos(2π fc t + θ2 ) with Y1 (t ) and Y2 (t ) independent then RZ (τ ) = SZ (f ) = 1 RY1 (τ ) + RY2 (τ ) 2 1 1 SY1 (f − fc ) + SY1 (f + fc ) + SY2 (f − fc ) + SY2 (f + fc ) 4 4 EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 115 / 130 Lecture Notes 5 Spectrum of Passband Signals Example: PSK For PSK x (t ) = ApT (t ). The spectrum is SZ (f ) = A2 T 4 EECS 455 (Univ. of Michigan) sinc2 (f − fc )T + sinc2 (f + fc )T . Fall 2012 September 6, 2012 116 / 130 Lecture Notes 5 Signal Space Concepts Orthogonal Expansions A set of real signals φ0 (t ), φ1 (t ), ..., φN −1 (t ) is said to be orthonormal over the interval [0, T ] if T φi (t )φj (t )dt = 0 EECS 455 (Univ. of Michigan) Fall 2012 0 i=j 1 i=j September 6, 2012 117 / 130 Lecture Notes 5 Signal Space Concepts The Gram-Schmidt Orthogonalization: Given a set of signals s0 (t ), s1 (t ), ..., sM −1 (t ) it is possible to find a set of orthogonal signals φ0 (t...
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