ch06-part 3

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Coherent Frequency-Shift Keying

M-ary PSK and M-ary QAM share a common property:Both are examples of linear modulation.

Coherent frequency-shift keying (FSK) is however, a

nonlinear method of passband data transmission.

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Binary FSK

In a binary FSK system, symbols 1 and 0 are distinguished fromeach other by transmitting one of two sinusoidal waves thatdiffer in frequency by a fixed amount.

A typical pair of sinusoidal waves is described by

where i= 1, 2, and Eb is the transmitted signal energy per bit;the transmitted frequency is

Thus symbol 1 is represented by s1(t), and symbol 0 by s2(t).

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Binary FSK

The FSK signal described here is a continuous-phase signal inthe sense that phase continuity is always maintained, including

the inter-bit switching times.

This form of digital modulation is an example of continuous-

phase frequency-shift keying (CPFSK).

Unlike coherent binary PSK, a coherent binary FSK system is

characterized by having a signal space that is two-dimensional

(i.e., N = 2) with two message points (i.e., M = 2), as shown in

Figure 6.25.

The two message points are defined by the

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Binary FSK

and

with the Euclidean distance between them equal to (2Eb).

Figure 6.25 also includes a couple of inserts, which showwaveforms representative of signals s1(t) and s2(t).

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Figure 6.25Signal-space diagram for binary FSK system. The diagram also

includes two inserts showing example waveforms of the twomodulated signals s1(t) and s2(t).

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Error Probability of Binary FSK

The average probability of bit error or, equivalently, the bit errorrate for coherent binary FSK is (assuming equiprobable

symbols)

We see that, in a coherent binary FSK system, we have to

double the bit energy-to-noise density ratio, Eb/N0, to maintain

the same bit error rate as in a coherent binary PSK system.

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Generation and Detection of Coherent Binary

FSK Signals

To generate a binary FSK signal, we may use the scheme shown

in Figure 6.26a.

The incoming binary data sequence is first applied to an on-off

level encoder, at the output of which symbol 1 is represented by

a constant amplitude of Eb volts and symbol 0 is representedby zero volts.

By using an inverter in the lower channel in Figure 6.26a, we in

effect make sure that when we have symbol 1 at the input, the

oscillator with frequency fl

in the upper channel is switched on

while the oscillator with frequency f2 in the lower channel is

switched off, with the result that frequency flis transmitted.

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Figure 6.26Block diagrams

for (a) binary FSK

transmitter and(b) coherent

binary FSK

receiver.

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FSK Signals

When we have symbol 0 at the input, the oscillator in the upper

channel is switched off and the oscillator in the lower channel is

switched on, with the result that frequency f2is transmitted.

The two frequencies f1 and f2 are chosen to equal different

integer multiples of the bit rate 1/Tb. In the transmitter of Figure 6.26a, we assume that the two

oscillators are synchronized, so that their outputs satisfy the

requirements of the two orthonormal basis function 1(t) and

2

(t).

To detect the original binary sequence given the noisy received

signal x(t), we may use the receiver shown in Figure 6.26b.

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FSK Signals

It consists of two correlators with a common input which are

supplied with locally generated coherent reference signals 1(t)

and 2(t).

The correlator outputs are then subtracted, one from the other,

and the resulting difference, y, is compared with a threshold ofzero volts.

lf y > 0, the receiver decides in favor of 1.

On the other hand, if y < 0, it decides in favor of 0.

lf y is exactly zero, the receiver makes a random guess in favorof 1 or 0.

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Power Spectra of Binary FSK Signals

Consider the case which the two transmitted frequencies f1and

f2 differ by an amount equal to the bit rate 1/Tb, and their

arithmetic mean equals the nominal carrier frequency fc; phase

continuity is always maintained, including inter-bit switching

times. We may express this special binary FSK signal as follows:

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Using a wellknown trigonometric identity, we get

The plus sign corresponds to transmitting symbol 0, and the

minus sign corresponds to transmitting symbol 1.

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It is apparent that the in-phase and quadrature components of

the binary FSK signal are independent of each other.

Accordingly, the baseband power spectral density of FSK signal

equals the sum of the power spectral densities of these two

components, as shown by

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Minimum Shift Keying

By proper use of the phase when performing detection, it is possible to

improve the noise performance of the receiver significantly.

This improvement is, however, achieved at the expense of increased

receiver complexity.

Consider a continuous-phase frequency-shift keying (CPFSK) signal,which is defined for the interval 0 tTbas follows:

where Eb is the transmitted signal energy per bit, and Tb is the bit

duration.

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The phase (0), denoting the value of the phase at time t = 0,sums up the past history of the modulation process up to time t =

0.

The frequencies f1and f2are sent in response to binary symbols

1 and 0 appearing at the modulator input, respectively. Another useful way of representing the CPFSK signal s(t) is to

express it in the conventional form of an angle-modulated signal

as follows:

where (t) is the phase of s(t).

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The phase (t) of a CPFSK signal increases or decreases linearly

with time during each bit duration of Tbseconds, as shown by

where the plus sign corresponds to sending symbol 1, and the

minus sign corresponds to sending symbol 0; parameter h is

given by

The nominal carrier frequency fcis obtained as

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Phase Trellis

We find that at time t= Tb,

That is to say, the sending of symbol 1 increases the phase of a

CPFSK signal s(t) by hradians, whereas the sending of symbol

0 reduces it by an equal amount.

The variation of phase (t) with time tfollows a path consisting

of a sequence of straight lines, the slopes of which represent

frequency changes. Figure 6.27 depicts possible paths starting from time t = 0.

A plot like that shown in Figure 6.27 is called a phase tree.

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Figure 6.27Phase tree.

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Phase Trellis

According to Figure 6.27 the phase change over one bit interval

is radians.

In contrast, we have a completely different situation when the

deviation ratio his assigned the special value of 1/2.

We now find that the phase can take on only the two values /2at odd multiples of Tb, and only the two values 0 and at even

multiples of Tb, as in Figure 6.28.

This second graph is called a phase trellis, since a "trellis" is a

treelike structure with remerging branches. Each path from left to right through the trellis of Figure 6.28

corresponds to a specific binary sequence input.

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Figure 6.28Phase trellis; boldfaced path represents the

sequence 1101000.

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Phase Trellis

For example, the path shown in boldface in Figure 6.28 corresponds to

the binary sequence 1101000 with (0) = 0.

Henceforth, we assume that h= 1/2.

With h = 1/2, we find that the frequency deviation (i.e., the difference

between the two signaling frequenciesf1and f2) equals half the bit rate. This is the minimum frequency spacing that allows the two FSK

signals representing symbols 1 and 0 to be coherently orthogonal in the

sense that they do not interfere with one another in the process of

detection.

It is for this reason that a CPFSK signal with a deviation ratio of onehalf is commonly referred to as minimum shift keying (MSK).

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