ELEC3027 Radio Communications Background Information on Amplitude Modulation

Transcription

1 ELEC327 Raio Communications Backgroun Information on Moulation 1 Analogue Moulation 1.1 Moulation When a Raio Frequency (RF) signal is place onto the antenna of a transmitter, it will propagate through free space an can be etecte on the antenna of a receiver. The higher the frequency of this signal, the smaller the antennas that are require. However, we are often intereste in communicating relatively low frequency message signals, such as auio. Hence, we must moulate our low frequency message signal onto a high frequency carrier, in orer to transmit it. This has the ae benefit of allowing us to moulate ifferent message signals onto ifferent carrier frequencies, in orer to transmit them without interfering with each other. Figure 1 shows the schematic of a transmission scheme that uses Moulation (AM) to transmit a time-varying input signal x(t), as well as emoulation to obtain a receive signal ˆx(t). Moulator A cos(2πf c t) Demoulator x(t) + y(t) u(t) LPF ˆx(t) Figure 1: AM moulation an emoulation Operation The AM moulator of Figure 1 uses the message signal x(t) to vary the amplitue of the carrier sinusoi cos(2πf c t), where the carrier frequency f c is usually much higher than the highest frequency in the message signal. As shown in Figure 1 y(t) = [A + x(t)] cos(2πf c t), (1) 1

2 where A is a constant DC offset. In the AM emoulator of Figure 1, the ioe symbol represents a rectifier which gives u(t) = { y(t) if y(t) > otherwise. (2) Finally, the Low Pass Filter (LPF) of Figure 1 is employe to provie the reconstructe message signal ˆx(t). In orer to show how the emoulator works, let s consier some examples in the next sections Example 1 Suppose that the message signal of Figure 1 is a simple sinewave x(t) = cos(2πf m t) having a frequency of f m = 2 khz, as shown in Figure 2. Note that in aition to the time omain plot of x(t), Figure 2 uses the Power Spectral Density (PSD) of x(t) to show it in the frequency omain. 2 Plot of x(t) Power spectral ensity of x(t) Power/frequency (B/hz) 2 4 Spike at f m =2 khz Frequency (MHz) Figure 2: Plot an PSD for the message signal x(t) = cos(2πf m t), where f m = 2 khz. Put simply, Fourier theory states that any message signal x(t) can be constructe from a sum of sinusois, having ifferent frequencies an various amplitues (as well as phases). Essentially, the PSD of Figure 2 shows the frequencies of those sinusois that have high amplitues. Since our message signal x(t) = cos(2πf m t) can be thought of as a sum of just a single f m = 2 khz sinusoi, the PSD of Figure 2 contains only a single spike at this frequency, showing that it is the only frequency that is associate with a high amplitue in x(t). 2

3 Using the signal x(t) = cos(2πf m t), Equation 1 becomes y(t) = [A + cos(2πf m t)] cos(2πf c t). (3) Using the trigonometric ientity cos(α) cos(β) = 1 cos(α β) + 1 cos(α + β), we obtain 2 2 y(t) = A cos(2πf c t) cos(2π[f c f m ]t) cos(2π[f c + f m ]t). (4) This shows that when x(t) = cos(2πf m t), the AM signal y(t) is the sum of three sinusois: one with a frequency equal to the ifference between the carrier an message frequencies [f c f m ]; one with a frequency equal to the carrier frequency f c ; one with a frequency equal to the sum of the carrier an message frequencies [f c + f m ]. Figure 3 provies the plot an PSD of the resultant moulate signal y(t) for the case where a carrier frequency of f c = 25 khz an a DC offset of A = 1.5 are use. Note that the signal x(t) acts as an envelope for the carrier sinusoi. Also note that in accorance with Fourier theory, the PSD of y(t) contains spikes at the three frequencies liste above, namely [f c f m ] = 23 khz, f c = 25 khz an [f c + f m ] = 27 khz. In other wors, each spike correspons to a ifferent term in Equation 4, as ientifie by the annotations in Figure 3. 4 Plot of y(t) 2 2 Power/frequency (B/hz) cos(2π[f c f m ]t) Acos(2πf c t) Power spectral ensity of y(t).5cos(2π[f c +f m ]t) f c =25 khz Frequency (MHz) Figure 3: Plot an PSD for the moulate signal y(t), for the case where the message signal x(t) of Figure 2 is moulate onto a carrier having a frequency of f c = 25 khz, using a DC offset of A = 1.5, as shown in Figure 1. 3

4 Figure 4 shows the corresponing signals that are obtaine by the emoulator of Figure 1. In accorance with Equation 2, the rectifier of Figure 1 clips the negative part of the moulate signal y(t) in orer to provie u(t). Following this, the reconstructe message signal ˆx(t) is obtaine by simply smoothing away the high frequency oscillations in the signal u(t). This is achieve by the LPF of Figure 1, which only passes the low frequency part of u(t) that correspons to the message signal x(t). As shown in Figure 4, the resultant reconstructe message signal ˆx(t) is the same as the message signal x(t) of Figure 2 (apart from a DC offset an a scaling factor, which are easily remeie). 4 Plot of u(t) Plot of ^x(t) Figure 4: Plots of the signals u(t) an ˆx(t), for the case where the moulate signal y(t) of Figure 3 is emoulate as shown in Figure Example 2 Let s now consier the case where the more intricate message signal x(t) of Figure 5 is transmitte using the AM scheme of Figure 1. This message signal can be thought of as comprising a sum of many sinusois, which have frequencies no greater than f max = 1 khz, as shown in the Power Spectral Density (PSD) plot of Figure 5. Figure 3 provies the plot an PSD of the resultant moulate signal y(t) for the case where a carrier frequency of f c = 25 khz an a DC offset of A = 1.5 are use to moulate the message signal x(t) of Figure 5. Again, the message signal x(t) can be seen to act as an envelope for the carrier sinusoi. As Equation 4 implies, for each of the many sinusois that comprise the message signal x(t), a pair of sinusois is generate in y(t), having frequencies on either sie of the carrier frequency f c. The group of sinusois having frequencies above the carrier frequency f c are referre to as 4

5 2 Plot of x(t) Power spectral ensity of x(t) Power/frequency (B/hz) f max =1 khz Frequency (MHz) Figure 5: Example plot an PSD for the message signal x(t) of Figure 1. 4 Plot of y(t) 2 V ppmax V ppmin Power spectral ensity of y(t) Power/frequency (B/hz) 2 Acos(2πf t) 4 c x(t)cos(2πf c t) 6 8 f c =25 khz Frequency (MHz) Figure 6: Plot an PSD for the moulate signal y(t), for the case where the message signal x(t) of Figure 5 is moulate onto a carrier having a frequency of f c = 25 khz, using a DC offset of A = 1.5, as shown in Figure 1. 5

6 the upper sieban. Similarly, the group having frequencies below f c are the lower sieban. It is these siebans that contain all of the message information. Note that the sinusoi at the carrier frequency f c oes not contain any message information. However, it is still useful since it helps the receiver to lock-on to the transmission. As in the first example, the emoulator of Figure 1 is able to reconstruct the message signal x(t), as shown in Figure 7. 4 Plot of u(t) Plot of ^x(t) Figure 7: Plots of the signals u(t) an ˆx(t), for the case where the moulate signal y(t) of Figure 6 is emoulate as shown in Figure 1. Observe in Figure 5 that the maximum an minimum amplitues of the signal x(t) are max[x(t)] 1.5 an min[x(t)] 1. As shown in Figure 6, these values affect the maximum an minimum peak-to-peak amplitues of the moulate signal y(t). Provie that the DC offset satisfies A > min[x(t)], the peak-to-peak amplitues are given by V ppmax = 2(A + max[x(t)]), (5) V ppmin = 2(A + min[x(t)]). (6) In the example of Figure 6, we obtain V ppmax = 6 an V ppmin = 1, since it employs a DC offset of A = 1.5. The moulation factor m of a moulate signal y(t) is efine as m = V ppmax V ppmin V ppmax + V ppmin. (7) Note that in cases where the DC offset A is large, V ppmax an V ppmin will have similar values an the moulation factor will be close to zero. As A is reuce towars min[x(t)], V ppmin will approach zero an the moulation factor m will increase towars one. Since the moulation 6

7 factor is in the range [, 1], it can be expresse as a percentage, whereupon it is calle the moulation inex. When the moulation inex is less than 1%, the AM signal y(t) is sai to be unermoulate. By contrast, the AM signal y(t) is sai to be 1% moulate when the moulation inex is 1%. Note that if the DC offset oes not satisfy A > min[x(t)], then the AM signal y(t) becomes overmoulate. In this case, the emoulator of Figure 1 is unable to reconstruct the message signal x(t) because the envelope of the moulate signal will cross-over itself, as shown for A =.5 in Figure 8. The result is that the parts of x(t) that o not satisfy A > x(t) become inverte in the reconstructe message signal ˆx(t), as shown in Figure 8. 4 Plot of y(t) Plot of ^x(t) Figure 8: Plots of the signals y(t) an ˆx(t), for the case where the message signal x(t) of Figure 5 is overmoulate using a DC offset of A =.5 as shown in Figure Double SieBan Suppresse Carrier Moulation In Section 1.1 we consiere AM, which as a DC offset A to the message signal x(t) in orer to facilitate a simple emoulator, comprising only a rectifier an an LPF. In aition to its simplicity, the emoulator of Section 1.1 has the avantage of being non-coherent, which means that it oes not require knowlege of the carrier frequency an phase in orer to perform emoulation. However, the AM scheme of Section 1.1 fails when the DC offset is not large enough to prevent overmoulation, as escribe in Section In this section we consier a moulation scheme that can successfully perform emoulation when the DC offset is not large enough to prevent overmoulation. In fact, the moulation scheme consiere in this section oes not require a DC offset at all, as shown in the schematic of Figure 9. Since removing the DC offset is equivalent to setting A =, Equation 4 implies 7

8 that the sinusoi at the carrier frequency f c is remove or suppresse from the moulate signal y(t), leaving only the two siebans. For this reason, the moulation scheme consiere in this section is calle Double SieBan Suppresse Carrier (DSBSC) moulation. Moulator cos(2πf c t) Demoulator 2 cos(2πf c t) x(t) y(t) u(t) LPF ˆx(t) Figure 9: DSBSC moulation an emoulation. The avantage of omitting the sinusoi at the carrier frequency is that this makes DSBSC much more power efficient than AM. This is because the sinusoi at the carrier frequency typically accounts for a large fraction of the transmit power in AM, but oes not carry any information about the message signal x(t). However, the isavantage is that coherent emoulation is require. More specifically, the emoulator is require to use carrier recovery techniques in orer to etermine the exact frequency an phase of the carrier sinusoi. These are require in orer to generate the signal 2 cos(2πf c t) that is use to perform emoulation in Figure 9. In Section 1.5, we will see what happens if the wrong frequency or phase are use for this signal Mathematics In orer to unerstan the operation of the DSBSC scheme shown in Figure 9, let s consier the associate mathematics. As shown in Figure 9 y(t) = x(t) cos(2πf c t), (8) u(t) = 2x(t) cos(2πf c t) cos(2πf c t). (9) Using the trigonometric ientity 2 cos(θ) cos(θ) = 1 + cos(2θ), we get u(t) = x(t) (1 + cos(4πf c t)), (1) = x(t) + x(t) cos(4πf c t). (11) The signal u(t) contains a high frequency component x(t) cos(4πf c t), which is filtere away by the LPF of Figure 9. As a result, the signal x(t) is reconstructe ˆx(t) = x(t). (12) Example Suppose that the message signal x(t) of Figure 5 is moulate onto an f c = 25 khz carrier, as shown in Figure 9. The resultant signal y(t) = x(t) cos(2πf c t) is obtaine by using x(t) to envelope the carrier wave cos(2πf c t), as shown in Figure 1. The PSD of y(t) is obtaine by convolving the ouble-sie PSD of x(t) with the PSD of the carrier wave, which resembles an impulse at f c = 25 khz. This convolution moves the ouble-sie PSD to the location of the impulse, as shown in Figure 1. 8

9 2 Plot of y(t) Power spectral ensity of y(t) Power/frequency (B/hz) B=2 khz f c =25 khz Frequency (MHz) Figure 1: Example plot an PSD for the moulate signal y(t) of Figure 9. Here, the ouble-sie PSD of x(t) can be obtaine by mirroring the single-sie PSD shown in Figure 1 about a frequency of Hz (an subtracting 3 B from all non-zero frequencies). This mirroring introuces negative frequencies. It may seem strange to use negative frequencies, but they re convenient because they make the escribe convolution work. The strangeness surrouning negative frequencies can be mitigate by consiering that cos(θ) = cos( θ). Hence, cos(2πft) =.5 cos(2πft) +.5 cos(2π( f)t). Therefore, any component of a signal can be thought of as having half its power at a positive frequency an the other half at the corresponing negative frequency. Note that halving the power is achieve by subtracting the 3 B that is mentione in brackets above. As shown in Figure 1, the banwith of the signal y(t) is B = 2 khz, where B = 2f max. (13) In the receiver of Figure 9, the signal u(t) = x(t)+x(t) cos(4πf c t) appears as shown in Figure 11. As shown in the PSD of Figure 11, the component x(t) cos(4πf c t) may be remove by the LPF of Figure 9, which requires a cutoff frequency of between 1 an 49 khz. 1.3 Quarature Moulation So far, we have only consiere the use of a cosine carrier wave. If we ha use a sine carrier wave in Section 1.2, we woul have obtaine very similar results. This is because the sine an cosine functions iffer only by a phase shift of π/2 raians, ie cos(θ) = sin(θ + π/2). As a result, sin(θ) = if cos(θ) = ±1 an cos(θ) = if sin(θ) = ±1. These results inicate that 9

10 4 Plot of u(t) Power spectral ensity of u(t) Power/frequency (B/hz) x(t) x(t)cos(4πf c t) Frequency (MHz) Figure 11: Example plot an PSD for the moulate signal u(t) of Figure 9. the cosine an sine functions are orthogonal to each other. This means that a signal x i (t) that is amplitue moulate onto a cosine carrier wave will not interfere with another signal x q (t) that is moulate onto a sine carrier wave having the same frequency f c. In this way, we can transmit two signals at once, which is useful for stereo auio for example. We refer to these signals as the in-phase signal x i (t) an the quarature-phase signal x q (t). The aitional presence of the quarature-phase signal gives Quarature Moulation (QAM) its name. A schematic for a QAM scheme is shown in Figure 12. Moulator Demoulator cos(2πf c t) 2 cos(2πf c t) x i (t) u i (t) LPF ˆx i (t) + y(t) x q (t) u q (t) LPF ˆx q (t) sin(2πf c t) 2 sin(2πf c t) Figure 12: QAM moulation an emoulation. 1

11 As shown in Figure 12 y(t) = x i (t) cos(2πf c t) + x q (t) sin(2πf c t), (14) u i (t) = 2x i (t) cos(2πf c t) cos(2πf c t) + 2x q (t) sin(2πf c t) cos(2πf c t), (15) u q (t) = 2x q (t) sin(2πf c t) sin(2πf c t) + 2x i (t) cos(2πf c t) sin(2πf c t). (16) Note that as in the DSBSC scheme shown in Figure 9, the QAM moulate signal y(t) of Figure 12 has a banwith of B = 2f max. The QAM scheme can therefore transmit ouble the amount of information in the same amount of banwith. It pays for this by requiring ouble the amount of transmit power, ouble the amount of harware an a more sophisticate mechanism for synchronising the receiver with the transmitter. Using the trigonometric ientities 2 cos(θ) cos(θ) = 1+cos(2θ), 2 sin(θ) sin(θ) = 1 cos(2θ) an 2 cos(θ) sin(θ) = sin(2θ), we get u i (t) = x i (t) + x i (t) cos(4πf c t) + x q (t) sin(4πf c t), (17) u q (t) = x q (t) x q (t) cos(4πf c t) + x i (t) sin(4πf c t). (18) After the high-frequency components of u i (t) an u q (t) are remove by the LPFs shown in Figure 12, we obtain as esire. ˆx i (t) = x i (t), (19) ˆx q (t) = x q (t), (2) Note that DSBSC can be thought of as a special case of QAM, in which x q (t) =. 1.4 Complex Quarature Moulation Note that just like how the QAM scheme of Figure 12 transmits a signal comprising the two components x i (t) an x q (t), there are two components to a complex number, namely the real part an the imaginary part. Complex numbers can therefore conveniently represent the two parts of our signal, accoring to where j = 1. x(t) = x i (t) + jx q (t), (21) Note that in a real circuit for a QAM moulator, we can t use an imaginary voltage to represent the quarature-phase component x q (t)! Remember that complex numbers are convenient; they allow us to simplify the mathematics of complicate QAM schemes, while remaining equivalent to them, as we ll see below. Inee, by using complex numbers, we can transform the (complicate) schematic of Figure 12 into the (simpler) one of Figure Mathematics As shown in Figure 13 y(t) = Re [ x(t)e j2πfct], (22) 11

12 Moulator Demoulator e j2πfct 2e j2πfct x(t) Re( ) y(t) u(t) LPF ˆx(t) Figure 13: Complex QAM moulation an emoulation. where Re[a + jb] = a. Using Euler s formula e jθ = cos(θ) j sin(θ) an j 2 = 1, we get y(t) = Re [x i (t) cos(2πf c t) jx i (t) sin(2πf c t) + jx q (t) cos(2πf c t) + x q (t) sin(2πf c t)], (23) = x i (t) cos(2πf c t) + x q (t) sin(2πf c t), (24) just like in Equation 14. Hence, the approaches of Figures 12 an 13 are equivalent. In the emoulator we have u(t) = 2y(t)e j2πfct. (25) Using Euler s formula e jθ = cos(θ) + j sin(θ) an the trigonometric prouct ientities of Section 1.3, we get u(t) = 2x i (t) cos(2πf c t) cos(2πf c t) + 2x q (t) sin(2πf c t) cos(2πf c t) + j2x q (t) sin(2πf c t) sin(2πf c t) + j2x i (t) cos(2πf c t) sin(2πf c t), (26) = x i (t) + x i (t) cos(4πf c t) + x q (t) sin(4πf c t) + jx q (t) jx q (t) cos(4πf c t) + jx i (t) sin(4πf c t). (27) After the high-frequency components of u(t) are remove by the LPF shown in Figure 13, we obtain as esire. ˆx(t) = x i (t) + jx q (t), (28) Phasors Note that the complex signal x(t) can be represente using a phasor. Here, x(t) = x i (t) + jx q (t) = x(t) e j x(t), (29) where the amplitue x(t) an phase x(t) are given by x(t) = x 2 i (t) + x2 q (t), (3) x(t) = arctan(x q (t)/x i (t)). (31) 12

13 Im[x(τ)] 1 x(τ) 2 1 x(τ) Re[x(τ)] -2-3 Figure 14: Phasor iagram for a complex signal x(τ) = 3 + 2j. For example, suppose that at a particular time instant where t = τ, the signal has a value of x(τ) = 3 + 2j. In this case x(τ) = 13 an x(τ) = 2.55 raians, as shown in the phasor iagram of Figure 14. Note that amplitue an phase can have a number of slightly ifferent meanings when consiering the phasors of sinusoial signals, like the moulate carrier y(t) in Figures 9, 12 an 13. This is because a sinusoi c(t) = A cos(2πft + θ) will always have a purely real value, irrespective of which time instant τ we pick for t. Hence, if we were to raw c(τ) = A cos(2πf τ + θ) in a phasor iagram, we woul always get a phase of c(τ) =, while the amplitue c(τ) = A cos(2πfτ + θ) woul epen on the particular value of τ an therefore be time varying. However, A cos(2πft + θ) = Re[Ae j(2πft+θ) ] = Re[Ae j2πft e jθ ]. Therefore, phasor iagrams are sometimes rawn for c (τ) = Ae j(2πfτ+θ), where c(t) = Re[c (t)]. Here, the phase c (τ) = 2πfτ + θ is time varying, while the amplitue c (τ) = A is constant. However, most frequently, phasor iagrams are rawn for c = Ae jθ, where c(t) = Re[c e j2πft ]. In this case, both the phase c = θ an the amplitue c = A are constant. Here, the phasor iagram is typically annotate with the sinusoi s frequency f, which is its thir parameter. 1.5 Carrier recovery As escribe in Section 1.2, the receiver is require to etermine the frequency an phase of the carrier before it can perform coherent etection, as employe in DSBSC an QAM schemes. This process is calle carrier recovery an it enables the receiver to generate the signal 2e j2πfct, as shown in Figure 13. However, in the case where carrier recovery fails, the receiver will generate the signal 2e j(2π[fc+f δ]t+θ δ ) = 2e j(2πfct+β) instea, where f δ is the frequency error, θ δ is the phase error an β = 2πf δ t + θ δ. 13

14 In this case, it can be shown that the signal recovere by the scheme of Figure 13 will be ˆx(t) = x i (t) cos(β) x q (t) sin(β) + jx q (t) cos(β) + jx i (t) sin(β). (32) By comparing Equations 21 an 32, we can see that phase an frequency errors cause the real part of ˆx(t) to become contaminate by x q (t). Likewise, the imaginary part of ˆx(t) becomes contaminate by x i (t). In other wors, the two signals x i (t) an x q (t) will interfere with each other. In fact, in cases where there is a π/2 phase error an no frequency error, we obtain β = θ δ = π/2 an ˆx(t) = x q (t) + jx i (t). (33) In this case, the signals x i (t) an x q (t) have swappe with each other! Furthermore, in cases where there is a frequency error f δ, the real an imaginary parts of ˆx(t) will resemble QAM signals. For example, when there is no phase error we obtain β = 2πf δ t an Re[ˆx(t)] = x i (t) cos(2πf δ t) x q (t) sin(2πf δ t), (34) which is similar to Equation 14. As a result, the recovere message signals will be shifte in the frequency omain from the baseban to become centere at f δ. These results show that owing to their coherent nature, it is vital for DSBSC an QAM schemes to successfully perform carrier recovery. 1.6 Channels with aitive complex noise So far, we ve assume that our channel oes not aversely affect our transmitte signal y(t). Let s see how our analogue moulation scheme performs when the channel imposes time-varying aitive complex noise n(t) = n i (t) + jn q (t). This may be use to simulate Aitive White Gaussian Noise (AWGN) by using a Gaussian istribution to ranomly select uncorrelate values for the real n i (t) an imaginary n q (t) parts of n(t). A channel having aitive complex noise is shown in Figure 15. Moulator e j2πfct n(t) Demoulator 2e j2πfct x(t) Re( ) y(t) + ŷ(t) u(t) LPF ˆx(t) Figure 15: Complex QAM moulation an emoulation when the channel has aitive complex noise. 14

15 1.6.1 Mathematics In this case, we have ŷ(t) = y(t) + n(t), (35) u(t) = 2ŷ(t)e j2πfct (36) = 2y(t)e j2πfct + 2n(t)e j2πfct. (37) Using Euler s formula e jθ = cos(θ) + j sin(θ) an the result for 2y(t)e j2πfct from Equation 27, we obtain u(t) = x i (t) + x i (t) cos(4πf c t) + x q (t) sin(4πf c t) + jx q (t) jx q (t) cos(4πf c t) + jx i (t) sin(4πf c t) + 2n i (t) cos(2πf c t) 2n q (t) sin(2πf c t) + j2n q (t) cos(2πf c t) + j2n i (t) sin(2πf c t). (38) At first glance it seems as if the LPF of Figure 15 will remove the various components of the noise n(t), since they has been shifte up to the carrier frequency by the multiplication with 2e j2πfct. However, if the noise is white (like AWGN) then it will affect all frequencies. As a result, the noise will still affect the base ban, even after the shift. Hence, the LPF of Figure 15 will only filter the noise, not remove it. The resultant recovere signal will therefore be ˆx(t) = x i (t) + jx q (t) + n i(t) + jn q(t) (39) = x(t) + n (t), (4) where n (t) is the shifte an filtere noise. This emonstrates that we can simply a the equivalent noise components n i (t) an n q (t) to the in-phase signal x i(t) an the quaraturephase signal x q (t), rather than going the whole-hog an simulating their moulation onto the channel. 1.7 Channels with a complex gain In aition to aitive complex noise, a channel can impose a complex gain a(t) = a i (t)+ja q (t) (which may vary with time). A complex channel gain can be use to simulate the path loss, slow faing an fast faing of narrowban channels. In the case of path loss, a purely-real value (i.e. a q (t) = ) that oes not vary with time is typically selecte for a(t) using the Hata moel. A purely-real constant value is also use in the case of slow faing. However, in this case, the value of a(t) is ranomly selecte from a lognormal istribution. In orer to evaluate the effect of this selection, we typically run a number of simulations, using a ifferent ranomly selecte value for a(t) in each. The performance observe in the various simulations can then be average to get an overall performance metric. In the case of fast faing, complex time-varying values are use for a(t). The real a i an imaginary a q parts of these values will be ranomly selecte using a Gaussian istribution, yieling magnitues having a Rician or Rayleigh istribution. Furthermore, a Doppler filter may be applie to inuce correlation. Of course, the prouct of the path loss, slow faing an fast faing channel gains can be use for a(t), when simulating a channel exhibiting all of these characteristics. A channel having a complex gain is shown in Figure

16 Moulator Demoulator e j2πfct a(t) 2e j2πfct x(t) Re( ) y(t) ŷ(t) u(t) LPF ˆx(t) Figure 16: Complex QAM moulation an emoulation when the channel has a complex gain Mathematics In this case, we have ŷ(t) = a(t)y(t). (41) Using Equation 24, we obtain ŷ(t) = a i (t)x i (t) cos(2πf c t) + a i (t)x q (t) sin(2πf c t) + ja q (t)x i (t) cos(2πf c t) + ja q (t)x q (t) sin(2πf c t), (42) Using Euler s formula e jθ = cos(θ) + j sin(θ) an the trigonometric prouct ientities of Section 1.3, we get u(t) = a i (t)x i (t) + a i (t)x i (t) cos(4πf c t) + a i (t)x q (t) sin(4πf c t) + ja q (t)x i (t) + ja q (t)x i (t) cos(4πf c t) + ja q (t)x q (t) sin(4πf c t) + ja i (t)x q (t) ja i (t)x q (t) cos(4πf c t) + ja i (t)x i (t) sin(4πf c t) a q (t)x q (t) + a q (t)x q (t) cos(4πf c t) a q (t)x i (t) sin(4πf c t). (43) After the high-frequency components of u(t) are remove by the LPF shown in Figure 16, we obtain ˆx(t) = a i (t)x i (t) + ja q (t)x i (t) + ja i (t)x q (t) a q (t)x q (t), (44) = (a i (t) + ja q (t))(x i (t) + jx q (t)), (45) = a(t)x(t), (46) which shows that the complex gain a(t) can be applie irectly to the complex signal x(t), without simulating its moulation onto the channel! The effect of the channel be evaluate by consiering a(t) an x(t) as phasors. When two phasors are multiplie together, the result is a phasor having an amplitue equal to the prouct of the two original amplitues, while the resultant phase is given by the sum of the original phases. Hence, ˆx(t) = a(t) x(t), (47) ˆx(t) = a(t) + x(t). (48) Therefore, the channel changes the amplitue an phase of the signal x(t). 16

17 1.7.2 Equivalence You may woner how we can use complex numbers (which have an imaginary component) for a real-life thing like the gain of a channel. Well, a complex gain is a convenient way of representing a channel that changes the amplitue an the phase of the transmitte signal y(t). Consier the case where the channel of Figure 12 (which oes not use complex numbers) replaces y(t) of Equation 14 with ŷ(t) = a(t) x i (t) cos(2πf c t a(t)) + a(t) x q (t) sin(2πf c t a(t)), (49) where a(t) = a 2 i (t) + a2 q (t), (5) a(t) = arctan(a q (t)/a i (t)). (51) In this case, the scheme of Figure 12 woul obtain the equivalent result to Equation 44, with which is the real part of Equation 44, as well as ˆx i (t) = a i (t)x i (t) a q (t)x q (t), (52) ˆx q (t) = a i (t)x q (t) + a q (t)x i (t), (53) which is the imaginary part of Equation 44. The ifference is that the mathematics woul have been even more complicate! Note that Equations 52 an 53 show that the effect of a channel having a complex gain is to cause x i (t) an x q (t) to interfere with eachother. This may be attribute to the phase ifference a(t) between the receive carrier an that generate locally in the receiver. Note that this is the same result as that obtaine when carrier recovery fails, as escribe in Section 1.5. In fact, successful carrier recovery can etect an rectify the phase ifference impose by the channel. Therefore, we are typically most intereste in the magnitue of the complex gain a(t). 1.8 Channels with a complex gain an aitive complex noise Wireless channels typically impose a complex gain an aitive complex noise. These channels can be moelle using the schematic of Figure 17. Moulator Demoulator e j2πfct a(t) n(t) 2e j2πfct x(t) Re( ) y(t) + ŷ(t) u(t) LPF ˆx(t) Figure 17: Complex QAM moulation an emoulation when the channel has a complex gain an aitive complex noise. As escribe in Section 1.7, carrier recovery techniques can be use to rectify the phase ifference impose by the channel s complex gain. Furthermore, amplification can be use to 17

18 compensate for the magnitue of the complex gain. However, this will also amplify any aitive complex noise, which cannot be easily mitigate. For this reason, noise is particularly etrimental in wireless channels. 2 Digital Moulation ary Quarature Moulation The previous section showe that analogue moulation schemes are susceptible to noise. In this section, we ll show that igital moulation schemes can achieve reliable communications even in the presence of relatively severe noise. The ifference between an analogue an a igital moulation scheme is the type of signal they are use to convey. As we showe in Section 1, analogue moulators transmit an analogue signal x(t). However, the analogue emoulator can never be sure if a particular component of the emoulate signal ˆx(t) is signal or noise. By contrast, igital moulators transmit igital signals, such as a sequence of binary igits b[n]. Since a bit can only have a value of or 1, the emoulator just has to choose from these two values when recovering the sequence ˆb[n]. While the emoulator can never be sure that it has mae the right choices, it will typically o a goo job so long as the noise is not really ba. We can construct a igital moulation scheme by converting the igital signal b[n] into an analogue signal x(t) an using the analogue moulation scheme of Figure 15. Once the emoulate signal ˆx(t) has been recovere, we just nee to convert it back into a igital signal ˆb[n]. Schematics for a Digital to Analogue Converter (DAC) an an Analogue to Digital Converter (ADC) are provie in Figures 18 an 19. b 1 [n] b[n] Serial to parallel 16QAM s[n] Impulse s(t) x(t) converter mapper generator LPF b 4 [n] Figure 18: Digital to analogue conversion using 16QAM. ˆx(t) Sampler ŝ[n] 16QAM emapper ˆb1 [n] ˆb4 [n] Parallel to serial converter ˆb[n] Figure 19: Analogue to igital conversion using 16QAM Serial to parallel conversion Consier the case where we wish to convey the bit sequence b[n] by sening k = 4 bits at a time. The first step is to convert our serial sequence of bits b[n] into four parallel sequences 18

19 b 1 [n], b 2 [n], b 3 [n] an b 4 [n], as shown in Figure 18. This is achieve by ecomposing b[n] into groups of four bits an istributing these among the four bit sequences b 1 [n], b 2 [n], b 3 [n] an b 4 [n]. For example, suppose that This gives {b[n]} 2 n=1 = [,, 1, 1, 1,, 1, 1,,, 1, 1,,,,,1, 1,1, 1]. (54) {b 1 [n]} 5 n=1 = [b[1], b[5], b[9], b[13], b[17]] = [, 1,,, 1], (55) {b 2 [n]} 5 n=1 = [b[2], b[6], b[1], b[14], b[18]] = [,,,, 1], (56) {b 3 [n]} 5 n=1 = [b[3], b[7], b[11], b[15], b[19]] = [1, 1, 1,, 1], (57) {b 4 [n]} 5 n=1 = [b[4], b[8], b[12], b[16], b[2]] = [1, 1, 1,, 1]. (58) Bit mapping The M = 16-ary Quarature Moulation (16QAM) bit mapper of Figure 18 converts the four bit sequences b 1 [n], b 2 [n], b 3 [n] an b 4 [n] into a single sequence of symbols s[n]. Since there are M = 2 k = 16 possible combinations of k = 4 bits, we require M = 16 ifferent values for the symbols of s[n]. Note that these values can be complex, since this is supporte by the moulator of Figure 15. We can therefore visualise the M = 16 ifferent symbol values as phasors in a constellation iagram. Figure 2 provies the 16QAM constellation iagram, in which M = 16 constellation points are arrange in a 4 4 gri. Im(s[n]) {b 1 [n],b 2 [n],b 3 [n],b 4 [n]} = Re(s[n]) Figure 2: 16QAM constellation iagram showing the Gray bit mapping. As shown in Figure 2, each of the M = 16 ifferent combinations of the four bits is mappe to a ifferent one of the M = 16 constellation points. Here, Gray bit mapping is employe, which ensures that the bit combinations that are mappe to neighbouring constellation points iffer only by one bit. The number of bits that iffer in a pair of bit combinations is calle their Hamming istance. 19

20 As shown in Figure 2, neighbouring constellation points are separate by a Eucliean istance of 2. The value chosen for affects the average transmit power. As shown in Figure 2, when {b 1 [n], b 2 [n], b 3 [n], b 4 [n]} has a value of 11, 111, 1111 or 111, s[n] = accoring to the Pythagorean theorem. By contrast, if {b 1 [n], b 2 [n], b 3 [n], b 4 [n]} has a value of 1, 1, 11, 11, 111, 111, 11 or 11, then s[n] = (3) Finally,, 1, 11 an 1 result in s[n] = (3) 2 + (3) 2. If we assume that all constellation points occur equally likely, then the average transmit power is given by E{ s[n] 2 } = 4/ / / = 1 2. If we want this to be unity, we can employ = 1/1. For the example b 1 [n], b 2 [n], b 3 [n] an b 4 [n] bit sequences of Equations 55 58, {s[n]} 5 n=1 = [ + 3j, 3j, + 3j, 3 + 3j, j]. (59) Impulse generation As shown in Figure 18, the next step is to convert our sequence of iscrete 16QAM symbols s[n] into a continuous function of time s(t). Suppose that we want to transmit our symbol sequence s[n] at a rate of f symbol = 1 symbols per secon. Each symbol therefore has a perio of t symbol = 1/f symbol =.1 ms. We can obtain a continuous function of time s(t) by generating impulses having the corresponing complex amplitues in the mile of each symbol perio. Figure 21 shows this for the example s[n] of Equation Plot of Re[s(t)] an Re[x(t)] Re[s(t)] Re[x(t)] Plot of Im[s(t)] an Im[x(t)] Im[s(t)] Im[x(t)] Figure 21: Plots of the real an imaginary parts of the signals s(t) an x(t) that correspons to the example s[n] of Equation 59. Here, x(t) has been obtaine by using the raise cosine filter characterise in Figure 22 to shape the pulses of s(t). 2

21 2.1.4 Pulse shaping As exemplifie in Figure 21, the signal s(t) changes very rapily. As a result, the maximum frequency f max in s(t) is very high. If we were to moulate s(t) on to the channel, then a very high banwith B woul result, since B = 2f max, as escribe in Section For this reason, we must apply a special type of LPF calle a Nyquist filter to s(t) before transmitting it, as shown in Figure 18. The frequency response of an example Nyquist filter is provie in Figure 22. This frequency response resembles half of a cosine cycle, that has been raise so that it is above the horizontal axis. For this reason, this LPF is calle a raise cosine filter. 1 Raise cosine filter frequency response, where α = Frequency / f symbol 1 Raise cosine filter impulse response, where α = 1 zero crossings zero crossings Time since impulse / t symbol Figure 22: Plots of the impulse an frequency response of a raise cosine filter having a roll-off factor of α = 1.. The LPF of Figure 18 is referre to as a pulse shaping filter, since it reshapes the impulses of s(t) so that they o not comprise any high frequency components. The manner in which the pulse shaping filter reshapes the impulses of s(t) is characterise by its impulse response. In aition to the frequency response of a pulse shaping filter, Figure 22 also provies the corresponing impulse response. In the scheme of Figure 18, the signal x(t) is obtaine by convolving the signal s(t) with the pulse shaping filter s impulse response. This replaces each impulse in the signal s(t) with a version of the pulse shaping filter s impulse response having the same amplitue an position in time. The signal x(t) is obtaine by summing all of the time-shifte impulse responses together at each moment in time. Figure 21 exemplifies this for the case of using the raise cosine filter characterise in Figure 22. Observe in Figure 21 that x(t) = s(t) whenever s(t) is impulse. This may seem surprising, since each point on x(t) is obtaine by summing together all of the time-shifte impulse responses, as escribe above. However, the impulse responses of raise cosine filters have zero crossings whenever the time since the impulse is a non-zero integer multiple of the symbol perio t symbol, 21

22 as shown in Figure 22. As a result, the value of x(t) at an instant when s(t) is impulse is affecte only by the corresponing impulse response; all of the other time-shifte impulse responses will be zero at this moment. This special feature of raise cosine filters means that they avoi Inter-Symbol Interference (ISI). In general, raise cosine filters have cut-off frequencies equal to half the symbol rate f symbol. They are parameterise by their roll-off factor α 1, which etermines the steepness of their frequency response The raise cosine filter that is characterise in Figure 22 employs the maximal value for its roll-off factor of α = 1. As a result, the amplitue of its frequency response graually changes from 1 to. This frequency response is calle the full-cosine roll-off characteristic. A steeper frequency response is exemplifie in Figure 23, in which α =.5. In general, the amplitue of a raise cosine filter s frequency response is unity for frequencies between an (1 α)f symbol /2, as shown in Figure 23. Furthermore, the frequency response resembles a raise cosine between frequencies of (1 α)f symbol /2 an (1 + α)f symbol /2. Note that the transition banwith is given by (1 + α)f symbol /2 (1 α)f symbol /2 = αf symbol. Finally, for frequencies above f max = (1 + α)f symbol /2, (6) the amplitue of a raise cosine filter s frequency response is zero, as shown in Figure 23. Here, f max is the highest frequency that will not be totally filtere out in x(t). Note that f max = 1 khz in the example x(t) of Figure 21, since f symbol = 1 khz an α = 1 in this case. 1 Raise cosine filter frequency response, where α =.5 (1 α)f symbol /2 (1+α)f symbol / Frequency / f symbol 1 Raise cosine filter impulse response, where α = Time since impulse / t symbol Figure 23: Plots of the impulse an frequency response of a raise cosine filter having a roll-off factor of α =.5. When the minimal value of α = is employe for the roll-off factor, an ieal Nyquist filter results. As shown in Figure 24, an ieal Nyquist filter has a brickwall frequency response. Note 22

23 that Figure 24 also provies the corresponing impulse response, which may be obtaine using the sinc function accoring to sinc(t/t symbol ), where sinc(x) = sin(πx)/(πx). 1 Raise cosine filter frequency response, where α = Frequency / f symbol 1 Raise cosine filter impulse response, where α = Time since impulse / t symbol Figure 24: Plots of the impulse an frequency response of an ieal Nyquist filter, which has a roll-off factor of α =. Accoring to Equation 13, the banwith B of an AM signal is ouble the maximum frequency f max present in the signal x(t). When a raise cosine filter is employe for pulse shaping, f max is given by Equation 6. Hence, the banwith require may be obtaine by combining Equations 13 an 6, yieling B = (1 + α)f symbol. (61) Therefore, low values of the roll-off factor α have the benefit of reucing the amount of banwith require B. However, Figures show that low roll-off factors are associate with impulse responses that take longer to ecay towars zero. As a result, impractically high orers are require in orer to implement raise cosine filters having low roll-off factors. Furthermore, when low roll-off factors are employe, the receiver is much more sensitive to offsets in its synchronisation with the transmitter Matche filters Suppose that the scheme of Figure 15 was employe to transmit the signal x(t) of Figure 21 over a severe AWGN channel. Also suppose that the LPF of Figure 15 employe a cut-off frequency of 25 khz. In this case, the reconstructe signal ˆx(t) that is provie to the ADC of Figure 19 woul look nothing like the transmitte signal x(t), as exemplifie in Figure 25. As escribe in Section 2.1.4, the signal x(t) comprises components having frequencies of up to f max = 1 khz. 23

24 However, the PSD of Figure 25 shows that the reconstructe signal ˆx(t) comprises components having frequencies of up to 25 khz, which is the cut-off frequency employe for the LPF of Figure 15. These components may be attribute to unfiltere AWGN. Clearly, some of the noise in the reconstructe signal ˆx(t) coul be remove by using a lower cut-off frequency for the LPF of Figure 15. However, if this cut-off frequency is reuce too far, then some of the esire signal x(t) may be filtere away too. 3 2 Plot of Re[^x(t)] an Re[x(t)] Re[^x(t)] Re[x(t)] Power spectral ensity of Re[^x(t)] Power/frequency (B/hz) x(t) AWGN Frequency (MHz) Figure 25: Plot an PSD of the real part of the moulate signal ˆx(t) from Figure 19 when matche filters are not employe. The solution is to use matche filters in the transmitter an receiver. More specifically, if the LPFs of Figures 18 an 15 have the same esign, then a maximum amount of AWGN can be remove, without filtering the esire signal away. The combination of the transmit an receive filters has a frequency response given by the prouct of their iniviual responses. We want this overall frequency response to be a raise cosine so that ISI can be avoie, as escribe in Section Therefore we shoul employ filters having frequency responses that are the square root of the raise cosine response. These filters are therefore calle root raise cosine filters. Returning to our example, consier the case where the transmit an receive filters are replace with root raise cosine filters having roll-off factors of α = 1. Using the same AWGN as in the example of Figure 25 in this case results in the reconstructe signal ˆx(t) shown Figure 26. Note that this much more closely resembles the transmitte signal x(t), emonstrating the benefit of matche filters. 24

25 3 2 2 Plot of Re[^x(t)] an Re[s(t)] Re[^x(t)] Re[s(t)] errors error Plot of Im[^x(t)] an Im[s(t)] ecision bounaries ecision bounaries Im[^x(t)] Im[s(t)] Figure 26: Plot of the real an imaginary parts of the moulate signal ˆx(t) from Figure 19 when matche filters are employe Decisions, ecisions, ecisions It is the job of the ADC shown in Figure 19 to consier the reconstructe signal ˆx(t) an ecie which bit values to output for b[n]. The first step is to sample ˆx(t) at the time instances where the signal s(t) was impulse in the transmitter. To help illustrate this, Figure 26 inclues plots of the real an imaginary parts of both ˆx(t) an s(t). The resultant samples ŝ[n] are then obtaine, as shown in Figure 19. For the example signal ˆx(t) of Figure 26, we get {ŝ[n]} 5 n=1 = [ j, j, j, j,.92.14j]. (62) These phasors are plotte in the 16QAM constellation iagram of Figure 27. The corresponing values for the bit sequences ˆb 1 [n], ˆb 2 [n], ˆb 3 [n] an ˆb 4 [n] are obtaine by selecting the constellation point that is nearest to each phasor. Figure 27 inclues ashe lines that ientify the regions in which each constellation point is the nearest. Note that these ecision bounaries are locate at 2, an 2 on both the real an imaginary axes. The corresponing ecision bounaries are inclue in the plot of ˆx(t) in Figure 26. In the case of the samples of Equation 62, we obtain the reconstructe bit sequences {ˆb 1 [n]} 5 n=1 = [, 1,,, 1], (63) {ˆb 2 [n]} 5 n=1 = [1,,,, 1], (64) {ˆb 3 [n]} 5 n=1 = [,, 1,, 1], (65) {ˆb 4 [n]} 5 n=1 = [1, 1, 1,, 1]. (66) 25

26 ŝ[3] {b 1 [n],b 2 [n],b 3 [n],b 4 [n]} = 1 11 Im(s[n]) 3 1 ŝ[4] 2 ŝ[1] ŝ[5] Re(s[n]) ŝ[2] 1 Figure 27: 16QAM constellation iagram showing the positions of the reconstructe symbols ŝ[n] of Equation 62. By comparing these bit sequences with those of Equations we can see that three bit errors have occurre. These may be attribute to the positioning of the samples in ŝ[n] within the wrong regions of Figure 27. Note that two bit errors have occurre, owing to the incorrect positioning of the sample ŝ[1], while ŝ[2] has cause only one bit error. This is because the noise has isplace ŝ[1] further than ŝ[2]. As escribe in Section 2.1.2, the Gray bit mapping of Figure 2 results in a Hamming istance of one between each pair of constellation points that are separate by the minimum Eucliean istance of 2a. Since AWGN channels are most likely to isplace the samples of ŝ[n] by a small Eucliean istance, Gray bit mapping results in a minimal BER. Note that ˆb 2 [n] an ˆb 4 [n] are more susceptible to bit errors than ˆb 1 [n] an ˆb 3 [n]. As shown in Figure 28, this is because every constellation point has a Eucliean istance of from the nearest ecision bounary for ˆb 2 [n] an ˆb 4 [n]. By contrast, some of the constellation points have a Eucliean istance of 3 from the ecision bounary for ˆb 1 [n] an ˆb 3 [n]. Therefore, more noise is require to corrupt these bits. Finally, the reconstructe bit sequence ˆb[n] is obtaine by performing the parallel to serial conversion of the sequences ˆb 1 [n], ˆb 2 [n], ˆb 3 [n] an ˆb 4 [n]. In the case of the sequences of Equations 63 66, we obtain the 2-bit sequence {ˆb[n]} 2 n=1 = [, 1,, 1, 1,,, 1,,, 1, 1,,,,,1, 1,1, 1], (67) 26

27 {b 1 [n],b 2 [n],b 3 [n],b 4 [n]} = ˆb2 [n] = ˆb1 [n] = Im(s[n]) = 2 Im(s[n]) = ˆb2 [n] = ˆb1 [n] = Im(s[n]) = 2 ˆb2 [n] = ˆb3 [n] = 1 ˆb3 [n] = ˆb4 [n] = ˆb4 [n] = 1 ˆb4 [n] = Re(s[n]) = Re(s[n]) = 2 Re(s[n]) = 2 Figure 28: Decision bounaries for Gray bit mappe 16QAM. which contains three bit errors, as escribe above. This correspons to a Bit Error Ratio (BER) of 3/2 = Eye iagrams In the previous section, the presence of noise in the channel cause a high BER to result. This effect can be analyse by consiering the corresponing eye iagram. Before we o this however, it is useful to raw an eye iagram for the case where there is no noise in the channel, for the sake of allowing a comparison. This eye iagram is shown in Figure 29. The eye iagram of Figure 29 was obtaine by first ecomposing the signal ˆx(t) into segments having urations equal to the symbol perio t symbol =.1 ms. Following this, the real an imaginary part of each pair of consecutive segments was plotte in Figure 29. The resultant figure is referre to as an eye iagram because it comprises empty regions that are shape like eyes. Note that in orer for these eye-shape regions to appear, the signal ˆx(t) must be long. For this reason, the.5 ms signal ˆx(t) that was use in the previous sections was extene to have a uration of 1.1 ms before the eye iagram of Figure 29 was rawn. Here, a uration of 1.1 ms can be ecompose into 11 segments having urations of t symbol =.1 ms. Hence, there are 1 possible pairings of consecutive segments an Figure 29 contains 1 plots. Observe that at the time instants labelle.5 ms an.15 ms in Figure 29, there are only 27

28 3 2 Eye iagram of Re[^x(t)] Eye iagram of Im[^x(t)] Figure 29: Eye iagram for 16QAM using a roll-off factor of α = 1 in the case where there is no noise in the channel. four possible amplitues for the real an imaginary parts of ˆx(t), namely 3,, an 3. This is because these time instants correspon to the miles of the consecutive symbol perios represente in Figure 29. At these time instants, ˆx(t) has a value equal to the corresponing symbol in s[n] when there is no noise in the channel, as escribe in the previous sections. Since the real an imaginary parts of s[n] can only take values of 3,, an 3 in 16QAM, these are the only amplitues that are possible for the real an imaginary parts of ˆx(t) at the time instants labelle.5 ms an.15 ms in Figure 29. Between the time instants labelle.5 ms an.15 ms in Figure 29, the real an imaginary parts of ˆx(t) can be seen to take sixteen ifferent paths, epening on the value of the two corresponing consecutive symbols in s[n]. Note that the paths have a non-zero with. This is because the path that ˆx(t) takes between two particular values of two corresponing consecutive symbols in s[n] also epens on the values of previous an subsequent symbols in s[n], owing to the tails in the impulse response of the pulse shaping filters, as escribe in Section As shown in Figure 3, this effect is exaggerate if the roll-off factor of the root raise cosine filters is reuce from α = 1 to α =.5, since this results in an impulse response having longer tails. The open-ness of the eye-shape regions in the eye iagram show how easy it is for the 16QAM emapper to make the correct ecisions. If the eyes are close, then it is ifficult for the emapper to make the correct ecisions, particularly if the sampler of Figure 19 is not perfectly synchronise with the impulse generator of Figure 18. For this reason, the BER is more sensitive to the synchronisation if a lower roll-off factor α is use, as escribe in Section This is the cost of using a lower banwith. 28

29 3 2 Eye iagram of Re[^x(t)] Eye iagram of Im[^x(t)] Figure 3: Eye iagram for 16QAM using a roll-off factor of α =.5 in the case where there is no noise in the channel. Returning to a roll-off factor of α = 1, the eye iagram of Figure 31 shows the effect of introucing a moerate amount of noise in the channel. Here, the eyes remain relatively open an a low BER coul be expecte. By contrast, Figure 32 provies an eye iagram for the case where the severe noise of Section is introuce by the channel. The close eyes shown in Figure 32 explain the high BER that was observe in Section Other igital moulation schemes In Section 2.1 we consiere 16QAM, which transmits k = 4 bits at a time by mapping each of the M = 2 k = 16 possible combinations of bits to a ifferent constellation point, in the particular manner shown in Figure 2. However in general, any number k of bits can be sent at once. Furthermore, any mapping of the M = 2 k ifferent combinations to the M = 2 k constellation points can be use. Finally, the M = 2 k constellation points can be positione anywhere in the complex plane. Of course, ifferent moulation schemes are associate with ifferent BER performances. Schematics for a general DAC an ADC are provie in Figures 33 an 34. Note that these are the same as those of Figures 18 an 19, with the exception that k can take any integer value. The ifferences between various igital moulation schemes are mae reaily apparent when their constellation iagrams are compare. Figures 35, 36 an 37 provie constellation iagrams for Binary (M = 2) Phase Shift Keying (BPSK), Quarternary (M = 4) Phase Shift Keying (QPSK) an M = 8-ary Phase Shift Keying (8PSK), respectively. 29