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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. Embed Size (px)
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