Bandpass Modulation and Demodulationpds7.egloos.com/pds/200805/13/35/04_Bandpass_Modulation_and... · 3 College of Information & Communications The world goes wireless! Prepared by

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College of Information & Communications

The world goes wireless! Prepared by Sung Ho Cho

Hanyang University

Bandpass Modulation and Demodulation

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Contents1. Digital Bandpass Modulation Techniques2. Detection of Signals in Gaussian Noise3. Coherent Detection4. Noncoherent Detection5. Complex Envelope6. Error Performance for Binary Systems7. M‐ary Signaling and Performance8. Symbol Error Probability for M‐ary Systems

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1. Digital Bandpass Modulation Techniques

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Digital Modulation

Definition:The process by which digital symbols are transformed into waveforms that are compatible with the characteristics of the channel.

In the baseband modulation: The waveforms normally take the form of shaped pulses.

In the bandpass modulation:The shaped pulses modulate a sinusoid, called a carrier wave.

We need to use a carrier for the radio transmission for the purpose of Reducing the size of the antenna.

The size of antenna depends on the wavelength λ and the application.Typically, the size of the antenna is λ/4, and λ = c/f (c = 3x108 m/s).For example, for a 900 MHz carrier, the equivalent antenna diameter would be about 8 cm.

Frequency‐division multiplexing

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Digital Bandpass Modulation

Definition:The process by which an information signal is converted to a sinusoidal waveform

The general form of carrier wave:

Two Basic Categories:Coherent Detection:

The receiver needs knowledge of carrier’s phase to detect the signals.The receiver is said to be phase locked to the incoming signal.

Noncoherent Detection: The receiver does not utilize such phase reference information.Phase estimation is not required.Reduced complexity, but increased error probability.

Time‐varying amplitude Carrier frequency Phase

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Digital Modulations (1/2)

Phase Shift Keying (PSK):

Frequency Shift Keying (FSK):Orthogonal signaling

Amplitude Shift Keying (ASK):

Amplitude Phase Keying (APK):

QAM if arranged in a rectangular constellation.

E: Symbol energyT: Symbol time duration

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Digital Modulations (2/2)

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2. Detection of Signals in Gaussian Noise

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Detection of Signals in Gaussian Noise

Two‐dimensional signal space with arbitrary equal‐amplitude vectors s1 and s2

The minimum error decision rule is equivalent to choosing the signal class such that is minimized.

Decision Rule:• Whenever the received signal r is located in Region 1, choose signal s1.

• Otherwise, choose signal s2.

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Correlation Receiver (1/3)

Received Signal:

Correlator Output:The correlator attempts to match the incoming received signal r(t) with each of the candidate prototype waveforms si(t), known a priori to the receiver.

Decision Rule:Choose the waveform si(t), that best matches or has the largest correlation with r(t).

Step 1: Compute a set of random variables formed at the output of the demodulator and sampled at time t = T such that

Step 2:

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Correlation Receiver (2/3)Correlator receiver with reference signals {si(t)}:

A bank ofM correlators

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Correlation Receiver (3/3)Correlator receiver with reference signals {ψj(t)}:

A bank of N correlators“Cost effective for MPSK”

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Example: Binary Correlator Receiver (1/2)

Using a Single Correlator:

Using Two Correlators:

Test Statistics (Gaussian random variable)

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Example: Binary Correlator Receiver (2/2)Binary Decision Threshold:

Likelihood of s1:

Likelihood of s2:

Minimum error criterion:

For equal energy, equally likely antipodal signal case:

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3. Coherent Detection

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Coherent Detection of MPSK (1/2)For typical coherent M‐ary PSK (MPSK) systems,

Transmitted signal:

Two basis functions:

Representation of the MPSK signal in terms of the basis functions:

Received signal:

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Coherent Detection of MPSK (2/2)

Demodulation for MPSK signals

In‐phase component

Quadrature component

Which region does it belong to?

Signal space and decision regions for a QPSK system

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Coherent Detection of MFSK (1/2)

For typical coherent M‐ary FSK (MFSK) systems,

Transmitted signal:

N (=M) basis functions:

Coefficients of the basis functions:

Distance between any two signal vectors for a given E:

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Coherent Detection of MFSK (2/2)

Partitioning the signal space for a 3‐ary FSK

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4. Noncoherent Detection

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Detection of Differential PSK (DPSK)Basic operations:

Differential encoding of the message sequence is required at the transmitter.The information is carried by the difference in phase between two successive waveforms.The carrier phase of the previous signaling interval is used as a phase reference for demodulation. In order to send i‐th message (i = 1, 2, …, M), the present signal waveform must have its phase advanced by φi=2πi/M (rad) over the previous waveform.The detector calculates the coordinates of the incoming signal by correlating it with locally generated sin and cos waveforms.The detector then measures the angle φi between the currently received signal vector and the previously received signal vector.

Signal space for DPSK:

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Example: Binary Differential PSKSequence of encoded bits c(k):

Binary message data sequence

2’s complement

Module‐2 addition

or

Differential encoding Differential detection

Recovery: ( ) ( ) ( )kckckm ⊕−= 1

Optimum detectionReference carrier

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Noncoherent Detection of FSK (1/2)Noncoherent detection of FSK waveforms:

Energy detectorwithout exploiting phase measurementsRequires twice as many channel branches as the coherent detector

Example: Quadrature receiver for noncoherent BFSK:

Quadrature receiver

Detection of signalwith frequency ω1

Detection of signalwith frequency ω2

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Noncoherent Detection of FSK (2/2)Noncoherent detection of FSK using envelope detectors:

The detectors are matched to the signal envelopes.The phase of the carrier is of no importance in defining the envelope.Looks functionally simpler than the quadrature receiver, but more expensive due to the existence of the analog bandpass filters.

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Tone Spacing for Noncoherent Orthogonal FSK (1/2)FSK is usually implemented as orthogonal signaling, but not all FSK signaling is orthogonal.

Tones f1 and f2 manifest orthogonality if, for a transmitted tone at f1, the sampled envelope of the receiver output filter tuned to f2 is zero (i.e., no crosstalk).

In order to ensure such orthogonality between tones in an FSK signaling set states that any pair of tones in the set must have a frequency separation that is a multiple of 1/T (Hz).

T = symbol duration

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Tone Spacing for Noncoherent Orthogonal FSK (2/2)Analytical description of the FSK tone:

Fourier transform of si(t):

The spectra of two such adjacent tones: Tone 1 with frequency f1Tone 2 with frequency f2

Required total bandwidth:(M+1)/THz

Minimum tone spacing for noncoherent orthogonal FSK signaling

Required total bandwidth

1 21f fT

− =

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Tone Spacing for Coherent Orthogonal FSKMinimum tone spacing for noncoherently detected orthogonal FSK signaling:

Minimum tone spacing for coherently detected orthogonal FSK signaling:

Coherently detected FSK can occupy less bandwidth than noncoherently detected FSK, while still retaining orthogonality.Coherent FSK is more bandwidth efficient.

1 21f fT

− =

1 21

2f f

T− =

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5. Complex Envelope

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Complex Representation of Bandpass Signals

Any real bandpass waveform s(t) can be represented as

Complex envelope g(t):Lowpass signalMagnitude & phase responses:

Therefore,

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6. Error Performance for Binary Systems

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Bit Error Probability for Coherent BPSKAssumption:

The signals are equally likely.Received signal:

Antipodal signals of the BPSK:

Decision rule:

Bit error probability, PB:

(where n(t) = AWGN)

1

2

2 00 2

b

b

a E

a EN

σ

=

= −

=

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Bit Error Probability for Coherent Orthogonal BFSKBit error probability, PB:

Here,ρ = cos θ = time cross‐correlation between s1(t) and s2(t) θ = angle between signal vectors s1 and s2For orthogonal BFSK, θ = π/2, and ρ = 0.

Therefore,

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Bit Error Probabilities for Selected Binary Modulations

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7. M‐ary Signaling and Performance

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Ideal Probability of Bit Error Performance

Ideal PB versus Eb/N0 curve over AWGN:

The Shannon limit represents the threshold Eb/N0 below which reliable communication cannot be maintained.

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M‐ary SignalingBit error probability for coherently detected M‐ary orthogonal signaling

Bit error probability for coherently detected M‐ary phase signaling:

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Vectorial View of MPSK Signaling

MPSK signal sets for M = 2, 4, 8, 16The error performance of MPSK signaling degrades as M increases.

Minimum noise vectorthat makes a symbol error

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QPSK Has the Same PB as BPSK.For a given PB, the general relationship between Eb/N0 and S/N:

S = average signal powerR = bit rate

The QPSK system can be characterized as two orthogonal BPSK channels.The QPSK bit stream is usually partitioned into an even and odd (I and Q) stream.Each new stream modulates an orthogonal component of the carrier at half the bit rate of the original stream. The I stream modulates the cos(ω0t) term, and the Q stream modulates the sin(ω0t) term.If the original QPSK vector has the magnitude of A, then each of the I and Q component vectors has the magnitude of 0.707A, and thus has half of the average signal power of the original QPSK signal.If the original QPSK waveform has a bit rate of R bits/s and an average power of Swatts, then the quadrature partitioning results in each of the BPSK waveforms having a bit rate of R/2 bits/s and an average power of S/2 watts.

Therefore, PB for QPSK = PB for BPSK

Symbol error probability for QPSK ≠ Symbol error probability for BPSKSymbol error probability for QPSK ≠ Symbol error probability for BPSK

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Vectorial View of MFSK Signaling

MFSK signal sets for M = 2, 3The signal from the orthogonal set is not particularly more vulnerable to a given noise vector as M increases.

Minimum noise vectorthat makes a symbol error

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PB for MFSK Improves as M Increases.For a given PB, the general relationship between Eb/N0 and S/N:

W = detection bandwidthR = bit rateT = symbol duration

(i.e., R = k times symbol rate 1/T)

For FSK signaling, W ≈ 1/T (symbol rate) or WT ≈ 1

When we use orthogonal signaling with symbols containing more bits, we need more power (i.e., more S/N), but the requirement per bit(i.e., Eb/N0) is reduced.

When we use orthogonal signaling with symbols containing more bits, we need more power (i.e., more S/N), but the requirement per bit(i.e., Eb/N0) is reduced.

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8. Symbol Error Probability for M‐ary Systems

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Symbol Error Probability, PE(M), for MPSK

PE(M) for coherent MPSK:Es = Eb(log2M)M = 2k

For large Es/N0 (energy‐to‐noise ratio ),

PE(M) for differentially coherent M‐ary DPSK:

For large Es/N0,

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Symbol Error Probability, PE(M), for MFSK

Upper bound of PE(M) for coherent MFSK:

PE(M) for noncoherent M‐aryorthogonal signaling:

For k > 7, there is almost no difference in performance between the two systems.

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PB versus PE for Orthogonal Signals

For equally likely orthogonal message symbols, Selecting any one of the (M‐1) erroneous symbol is equally likely.PB is always less than or equal to PE.

Example: Suppose “0 1 1” is transmitted for k=3.There are 2k‐1 = 7 ways of making a symbol error. Just because a symbol error is made does notmean that all the bits within the symbol will be in error.There are 2k/2 = 2k‐1 = 4 ways of making a bit error.Therefore, PB/PE = 4/7.

Symbol errors are equally likely.

Symbol errors are equally likely.

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PB versus PE for MPSK Signals

For equally likely MPSK message symbols, Each signal vector is not equi‐distant from all of the others.PB is always less than or equal to PE.

Example: Suppose “0 1 1” is transmitted for k=3.

Utilizing the Gray code, we haveFor BPSK, PE = PBFor QPSK, PE = 2PB

Symbol errors are not equally likely.

Symbol errors are not equally likely.

Adjacent symbols differ from one another in only one bit position.

Adjacent symbols differ from one another in only one bit position.

Binary code Gray code

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Bandpass Modulation and Demodulationpds7.egloos.com/pds/200805/13/35/04_Bandpass_Modulation_and... · 3 College of Information & Communications The world goes wireless! Prepared by