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Digital Communications: Introduction

Ref.: Communication Systems by A. Bruce Carlson and Paul Crilly 2010

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Sampling & Analog Pulse Modulation

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An electric signal satisfying certain requirements can be reproduced from an appropriate set of instantaneous samples. Sampling therefore makes it possible to transmit a message in the form of pulse modulation, rather than a continuous signal. Usually the pulses are quite short compared to the time between them, so a pulse modulated wave has the property of being “off” most of the time.

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Second, the time interval between pulses can be filed with sample values from other signals, a process called time-division multiplexing (TDM).

Pulse modulation offers two potential advantages over CW modulation. The transmitted power can be concentrated into short bursts instead of being generated continuously.

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But pulse modulation has the disadvantage of requiring very large transmission bandwidth compared to the message bandwidth. Consequently, the methods of analog pulse modulation discussed in this chapter are used primarily as message processing for TDM and/or prior to CW modulation.

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SAMPLING THEORY

Switching sampler (a) functional (b) waveforms (c) circuit

Chopper (Practical) Sampling

unipolar chopping

7Sampling as multiplication

Functional diagram

Switching function

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x(t) = Input analog signal

S(t) = Switching function

Xs(t ) = Sampled signal

Sampling rate fs == 1/Ts Hz

Since s(t) is periodic, it can be written as a Fourier series

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Input message spectrum

Output spectrum

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Spectra for switching sampling: (a) message; (b) properly sampled message, fs ˃2W; (c) undersampled aliased message, fs˂ 2W.

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The sampling operation has left the message spectrum intact, merely repeating it periodically in the frequency domain with a spacing of fs. The first term of the spectrum equation is precisely the message spectrum,attenuated by the duty cycle c0 = fsτ = τ /Ts.

Two conditions obviously are necessary to prevent overlapping spectral bands: the message must be bandlimited, and the sampling frequency must be sufficiently great that fs – W ≥W.

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The minimum sampling frequency

Nyquist rate

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Example: Bipolar Chopper

S(t) is a square wave alternating between 1 and 1.

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The Fourier transform of a bipolar square wave contains only only the odd harmonics of fs.

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Note that Xs(f) contains no DC component and only the odd harmonics of fs. Clearly, we can’t recover x(t) by lowpass filtering. Instead, the practical applications of bipolar choppers involve bandpass filtering. If we apply xs(t) to a BPF centered at some odd harmonic nfs, the output will be proportional to x(t) cos(nωst), a double-sideband suppressed -carrier waveform Thus, a bipolar chopper serves as a balanced modulator.

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Ideal SamplingThe ideal sampling function is a train of impulses

Instantaneous sampling

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The sampled wave becomes a train of impulses whose areas equal the instantaneous sample values of the input signal.

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and likewise

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Summary Theorem of uniform (periodic) sampling

If a signal contains no frequency components for |f|≥W, it is completely described by instantaneous sample values uniformly spaced in time with period Ts ≤1/2W. If a signal has been sampled at the Nyquist rate or greater (fs ≥ 2W) and the sample values are represented as weighted impulses, the signal can be exactly reconstructed from its samples by an ideal LPF of bandwidthB,

where W ≤B ≤ fs – W.

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PULSE-AMPLITUDE MODULATIONThe pulse amplitude varies in direct proportion to the sample values of x(t).

The output of the sampler xs(t) is PAM signals

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Flat-Top Sampling and PAM

PAM waveform obtained by the sample/hold ( S/H) technique

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Transmission bandwidth

PAM requires more transmission bandwidth AM CW modulation

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PULSE-TIME MODULATION

• Pulse-duration (PDM)(also called) Pulse-width modulation (PWM)

Pulse-position modulation (PPM)Time parameter of the pulse is being modulated, and the pulses have constant Amplitude. The pulse width or pulse position varies in direct proportion to the sample values of x(t).

26Types of pulse-time modulation.

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The duration of the kth pulse in the PDM signal

in which the unmodulated duration t0 represents x(kTs)= 0 and the modulation index µ controls the amount of duration modulation.

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in which the unmodulated position kTs + td represents x(kTs) 0 and the constant t0 controls the displacement of the modulated pulse.

The kth pulse in a PPM signal begins at time

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Generation of PDM or PPM: (a) block diagram; (b) waveforms.

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The system employs a comparator and a sawtooth-wave generator with period Ts. The output of the comparator is zero except when the message waveform x(t) exceeds the sawtooth wave, in which case the output is a positive constant A. Hence, the comparator produces a PDM signal with trailing-edge modulation of the pulse duration. (Reversing the sawtooth results in leading-edge modulation while replacing the sawtooth with a triangular wave results in modulation on both edges.) Position modulation is obtained by applying the PDM signal to a monostable pulse generator that triggers on trailing edges at its input and produces short output pulses of fixed duration.

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Pulse Code Modulation (PCM)

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PCM is a digital transmission system with an analog-to-digital converter (ADC) at the input and a digital-to-analog converter (DAC) at the output.

PCM Generation and ReconstructionGeneration system: The analog input waveform x(t) is lowpass filtered and sampled to obtain x(kTs ) .

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A quantizer rounds off the sample values to the nearest discrete value in a set of q quantum levels. The resulting quantized samples xq(kTs ) are discrete in time (by virtue of sampling) and discrete in amplitude (by virtue of quantizing).

PCM generation system;

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Let the analog message be a voltage waveform normalized such that │x(t)│ ≤ 1 VUniform quantization subdivides the 2-V peak-to-peak range into q equal steps of height 2/q V

The quantum levels are then taken to be at

q is an even integerThe normalized quantization-level step size

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Quantization characteristic.

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Next, an encoder translates the quantized samples into digital code words. The encoder works with M-ary digits and produces for each sample a codeword consisting of v digits in parallel. Since there are M˄v possible M-ary codewords with v digits per word, unique encoding of the q different quantum levels requires that M˄v≥ q. The parameters M, v, and q should be chosen to satisfy the equality, so that

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Each encoded sample is represented by a v-digit output word, so the signaling rate becomes r = vfs with fs ≥ 2W . Therefore, the bandwidth needed for PCMbaseband transmission is

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PCM receiverThe received signal may be contaminated by noise, but regeneration yields a clean and nearly errorless waveform if the signal-to-noise ratio (S/N)R is sufficiently large. The DAC operations of serial-to-parallel conversion, M-ary decoding, and sample-and-hold generate the analog waveform xq(t). This waveform is a “staircase” approximation of x(t), similar to flat-top sampling except that the sample values have been quantized. Lowpass filtering then produces the smoothed output signal , which differs from the message x(t) to the extent that the quantized samples differ from the exact sample values x(kTs) .

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Perfect message reconstruction is impossible in PCM, even when random noise has no effect. The ADC operation at the transmitter introduces permanent errors that appearat the receiver as quantization noise in the reconstructed signal.

PCM receiver

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reconstructed waveform. (low pass filtered)

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Bandpass Digital

Transmission

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DIGITAL CW MODULATIONA digital signal can modulate the amplitude, frequency, or phase of a sinusoidal carrier wave. If the modulating waveform consists of NRZ rectangular pulses, then the modulated parameter will be switched or keyed from one discrete value to another.ExampleBinary amplitude-shift keying (ASK) Binary frequency-shiftkeying (FSK) Phase-shift keying (PSK)

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BASK

BFSK

BPSK

Digital bits = 1 0 1 1 0 1 0 0Tb= bit period = 1/bite rate

Binary Signaling

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4-ary Signalling

s

ASK

PSK

FSK

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)2sin()( ftVtv•If the amplitude, V of the carrier is varied proportional to the information signal, a digital modulated signal is called Amplitude Shift Keying (ASK)•If the frequency, f of the carrier is varied proportional to the information signal, a digital modulated signal is called Frequency Shift Keying (FSK)

Carrier Signal

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•If the phase, θ of the carrier is varied proportional to the information signal, a digital modulated signal is called Phase Shift Keying (PSK)•If both the amplitude,V and the phase, θ of the carrier are varied proportional to the information signal, a digital modulated signal is called Quadrature Amplitude Modulation (QAM)

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M-ary Encoding• It is often advantageous to encode at a

level higher than binary where there are more then two conditions possible.

• The number of bits necessary to produce a given number of conditions is expressed mathematically as

N = number of bits necessary M = number of symbols, level or combinations possible with N bits.

MN 2log

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• The rate of change of a signal on the transmission medium after encoding and modulation have occurred.

baud = 1/ts

baud = symbol rate (symbol per second)

ts = time of one signaling element (symbol time (seconds)

Baud

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Passband digital modulation has form

Bits encoded in amplitude An, phase θn, or frequency θn=2π(fn-fc)t, which are constant over a bit time Tb.

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CONSTELLATION DIAGRAMGraphical representation of the complex envelope of each possible symbol state.

The x-axis represents the in-phase component and the y-axis the quadrature component of the complex envelope.

Thee distance between signals on a constellation diagram relates to how different the modulation waveforms are and how easily a receiver can differentiate between them.

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Amplitude Shift Modulation

•The binary ASK waveform be generated simply by turning the carrier on and off, a process described as on-off keying (OOK).

•M-ary ASK waveform has M-1 discrete “on’’ amplitudes as well as the “off’’.

M-ary Signaling M symbols (states)Logics 0, 1, 2, 3. …, M-1 States

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Each symbol carries (Log2 M) bits

M-ary ASK signal represents binary data at rate

r = Symbol rate rb= bite rateThe estimated transmission bit BT≈ r

BT ≈ rb/(Log2 M)

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spectral efficiency

Bps/Hz

Binary OOK: spectral efficiency =1bps/HzBinary ASK

Xc(t) = Ac cos(wct) logic 1

= 0 logic 0

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BPSK

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BASK modulator

Coherent demodulator

BASK

m(t) = Binary baseband data representing logics 1 or 0

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Detectors for BASK: Coherent Receiver

Coherent detection requires the phase information A coherent detector mixes the incoming signal with a

locally generated carrier reference Multiplying the received signal r(t) by the receiver

local oscillator (say Accos(wct)) yields a signal with a baseband component plus a component at 2fc

Passing this signal through a low pass filter eliminates the high frequency component

The output of the LPF is sampled once per bit period T

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Noncoherent ReceiverDoes not require a phase reference at the receiver If we do not know the phase and frequency of the carrier, we can use a noncoherent receiver to recover ASK signal

Envelope Detector

The simplest implementation of an envelope detector comprises a diode rectifier and smoothing filter. fo is the carrier frequency

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Phase-shift keying (PSK)The phase of the carrier signal is switched between 2 (for BPSK) or more (for MPSK) in response to the baseband digital dataThe information is contained in the instantaneous phase of the modulated carrierUsually this phase is imposed and measured with respect to a fixed carrier of known phase – Coherent PSKFor binary PSK, phase states of 0o and 180o are used

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BPSK Waveform

Xc(t)

It display antipodal signalling. I.e. symbols are equal and Opposite to each other, unlike ASK

Constellation Diagram

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BPSK - Implementation

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Xc(t)= Ac cos (ωc t + Øi) 0≤t≤Ts i=1, 2, …,M

MiMiti ,....1)1(2)(

In MPSK, the phase of the carrier takes on one of M possible values

M-ary PSK (MPSK)

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...........161688

422

PSKPSK

QPSKBPSKMPSKM k

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Quadrature PSK (QPSK)• Two BPSK in phase quadrature• QPSK (or 4PSK) is a modulation technique

that transmits 2-bit of information using 4 states of phases

• For example

2-bit Information

ø

00 001 π/210 π11 3π/2

Each symbol corresponds to two bits

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scQPSK TtiMitfActX

04,3,2,1,)1(22cos)(

scQPSK TtiMitActx

04,3,2,1,

4)1(2cos)(

We can also have

Øi = 45⁰, 135⁰, 225⁰, or 315⁰

scQPSK TtiMitActx

04,3,2,1,

4)1(2cos)(

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PSK signal constellations: (a) M =4; (b) M= 8.

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Frequency -Shift Keying

The instantaneous frequency of the carrier signal is switched between two (or more) values by the modulating digital data signal.

BFSK

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Generation of BFSK

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There are two basic methods for generation digital frequency modulation FSK : The digital signal x(t) controls a switch that selects the modulated frequency from a bank of M oscillators. The modulated signal is discontinuous at every switching instant. The resultant output spectrum will contain relatively large sidelobes which don’t carry any additional information and thus waste bandwidth. Discontinuities are avoided in continuous-phase FSK (CPFSK) where x(t) modulates the frequency of a single oscillator.

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fc1 fcM

Digital frequency modulation: (a) FSK; (b) continuous-phase FSK. The digital signal has M logic states (0, 1, 2, …, M-1)

fc2

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Quadrature Amplitude Modulation (QAM)

It combines amplitude and phase modulation

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Quadrature phase (Q)

In phase(I)

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Modulation Bits per symbol Symbol Rate

BPSK 1 1 x bit rate QPSK 2 1/2 bit rate 8PSK 3 1/3 bit rate 16QAM 4 1/4 bit rate 32QAM 5 1/5 bit rate 64QAM 6 1/6 bit rate