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Phase Shift Keying · Phase Shift Keying • Transmit information by modulating the phase of a sine...

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  • rem:Home:ese488:Lectures:4_QPSK:Lecture4b_QPSK.doc Page 1 of 10

    Phase Shift Keying

    • Transmit information by modulating the phase of a sine wave • Binary Phase Shift Keying (BPSK) - 180° Phase Shift – Cosine Channel Only

    • Quaternary Phase Shift Keying (QPSK) - 180° Phase Shift – Sine and Cosine Channels.

    Bit0 (T seconds) Bit1 (T seconds)

  • rem:Home:ese488:Lectures:4_QPSK:Lecture4b_QPSK.doc Page 2 of 10

    Receiver Block diagram for a BPSK receiver: s(t) Iin(t) Iout(t) y(t) n(t) (Local Oscillator)

    • Assume Coherence i.e. Local Oscillator is synchronized to s(t) • n(t) is White Gaussian Noise (WGN), ,

    Power Spectral Density

    N0 (W/Hz). • At the Integrator Input:

    • At the Integrator Output:

    • So the Integrator Output at time T, , is + Noise. What is the

    Standard Deviation ( ) for the Noise?

    + x

    t=(n+1)T

    + -

  • rem:Home:ese488:Lectures:4_QPSK:Lecture4b_QPSK.doc Page 3 of 10

    σout2 ≡Variance = E(N 2) − E 2(N) = E(N 2)

    since N is zero mean.

    σout2 = E n t( )cos(2πf0t)dt

    0

    T

    ∫⎡

    ⎣ ⎢

    ⎦ ⎥

    2⎛

    ⎝ ⎜ ⎜

    ⎠ ⎟ ⎟

    = E n t( )n u( )cos(2πf0t)cos(2πf0u)dudt0

    T

    ∫0

    T

    ∫⎛

    ⎝ ⎜

    ⎠ ⎟

    = E n t( )n u( )[ ]cos(2πf0t)cos(2πf0u)dudt0

    T

    ∫0

    T

    = N02δ(t − u)cos(2πf0t)cos(2πf0u)dudt

    0

    T

    ∫0

    T

    = N02

    cos2(2πf0t)dt0

    T

    = N04

    1+ cos(4πf0t)( )dt0

    T

    = N0T4

    σout = σout2 =

    12

    N0T

  • rem:Home:ese488:Lectures:4_QPSK:Lecture4b_QPSK.doc Page 4 of 10

    Waveforms for (all examples have Fs = 1000 Hz, BW = 500 Hz):

    Zoom In of Final Value of Integrator

  • rem:Home:ese488:Lectures:4_QPSK:Lecture4b_QPSK.doc Page 5 of 10

    Waveforms for:

    f0 =10 Hz, N0 =158µW /Hz, A = +1 V, and T =1 sec

    Zoom In of Final Value of Integrator

  • rem:Home:ese488:Lectures:4_QPSK:Lecture4b_QPSK.doc Page 6 of 10

    Histogram of the final values of Iout(T)

    o

    o

    o

    o

    Pr(Error) =12erfc

    AT2

    σout 2

    ⎜ ⎜ ⎜

    ⎟ ⎟ ⎟

    =12erfc

    AT2

    12

    N0T 2

    ⎜ ⎜ ⎜

    ⎟ ⎟ ⎟

    =12erfc A T

    2N0

    ⎝ ⎜

    ⎠ ⎟

    Pr(Error) =12erfc Eb

    N0

    ⎝ ⎜

    ⎠ ⎟

    Pr(0/1) | Pr(1/0)

  • rem:Home:ese488:Lectures:4_QPSK:Lecture4b_QPSK.doc Page 7 of 10

    The following graph is a plot of

    Pr(Error) vs EbN0

    where is the Energy per Bit and

    N0is the Power Spectral Density of the AWGN noise of the channel. For Example,

    N0 = 0.198W /Hz, A =1, T =1, Fs =100

    Eb =A2T2

    10log EbN0

    ⎝ ⎜

    ⎠ ⎟ =10log

    0.50.198⎛

    ⎝ ⎜

    ⎠ ⎟ = 4 dB

    Pr(Error) =12erfc Eb

    N0

    ⎝ ⎜

    ⎠ ⎟ =

    12erfc(1.58) = 0.0125

    This is a point on the curve below for Coherent PSK.

  • rem:Home:ese488:Lectures:4_QPSK:Lecture4b_QPSK.doc Page 8 of 10

  • rem:Home:ese488:Lectures:4_QPSK:Lecture4b_QPSK.doc Page 9 of 10

    Quaternary Phase Shift Keying

    • Use the Sine and Cosine channels simultaneously.

    • Transmit twice as much information in the same bandwidth! • Also uses twice the power

    o

    Icos_in(t) Icos_out(t) s(t) (Local Oscillator) n(t) Isin_in(t) Isin_out(t) (Local Oscillator)

    +

    x

    t=(n+1)T

    + -

    t=(n+1)T

    + -

    x

  • rem:Home:ese488:Lectures:4_QPSK:Lecture4b_QPSK.doc Page 10 of 10

    Cosine Channel :

    Icos_ in t( ) = s t( ) + n t( )[ ]cos(2πf0t)

    Icos_ out t( ) = ±Acos(2πf0t) + ± Asin(2πf0t) + n t( )[ ]cos(2πf0t)dt0

    T

    = ±Acos2(2πf0t)0

    T

    ∫ dt + ±Asin(2πf0t)cos(2πf0t)dt + n t( )cos(2πf0t)dt0

    T

    ∫0

    T

    = ± AT2

    + 0 + N

    σ cos_ out =12

    N0T

    Sine Channel :

    Isin_ in t( ) = s t( ) + n t( )[ ]sin(2πf0t)

    Isin_ out t( ) = ±Acos(2πf0t) + ± Asin(2πf0t) + n t( )[ ]sin(2πf0t)dt0

    T

    = ±Asin2(2πf0t)0

    T

    ∫ dt + ±Asin(2πf0t)cos(2πf0t)dt + n t( )sin(2πf0t)dt0

    T

    ∫0

    T

    = ± AT2

    + 0 + N

    σ sin_ out =12

    N0T

    References: • “Principles of Communications, Systems, Modulation, and Noise”, Ziemer and Tranter, pp.

    343, 455. • Dr. Morley’s EE437 Lecture Notes, Fall 2003.

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