Communication by Sampling
3.2 INFORMATION TRANSFER IN PULSE MODULATION SYSTEMS
It is instructive to compare the various pulse modulation systems in terms of infor-mation transfer. We use the Hartley-Shannon law,2 which states that if a transmission path is disturbed by white gaussian noise of power N, then the maximum information I in bits per second that can be conveyed accurately by the transmission path is given by
1= B /Og2 ( 1
+~)
(3.1)where B is the bandwidth of the channel, in Hz, and S is the power of the transmitted signal. In a real transmission there will always be some noise present; that is, N will have a finite value. Thus, the maximum information I about a message signal that can be conveyed by a transmission path is always limited.
In PAM the message is transmitted using pulses that can assume any amplitude value, using in the ideaP case a bandwidth equivalent to the message signal. However, although it is possible to generate an infinite number of amplitude values, the process of transmission will for a real system (that is, one that contains noise) make it impossi-ble to recover the original infinite amount of information. The maximum amount of information that can be transmitted will always be I as defined by Eq. (3.1).
Thus information about the message signal will be lost during the process of transmis-sion in a proportion dependent on the noise within the system. For this reason PAM is not considered for transmission purposes.
2 C. E. Shannon, "Communication in the presence of noise," Proceedings of the Institute of Radio Engi-neers, vol. 37, 1949, pp. 10-21.
3lt is assumed here that the received message, retrieved from the PAM signal, is obtained using a rectangular characteristic low-pass filter.
38 Pulse-Code Modulation
In PCM each amplitude value is approximated to the nearest permitted level, which is represented by a set of pulses. Suppose that there are 16 possible (quantized) levels defined in a given system, and that the message signal of maximum frequency 1m is sampled at 21m. In this case the 16 possible levels may be represented by a sequence of 4 pulses, and the total information to be transmitted is 81m. This informa-tion is finite, and can be better communicated via a transmission path at a compara-tively low to-noise ratio (see the discussion of error probability versus signal-to-noise ratio in Secs. 8.4.3 and 8.4.4).
With PCM the degradation that occurs in the complete transmission system is in the process of approximating the measured amplitude value. This is known as quanti-zation noise, and can be reduced by increasing the number of permitted levels. It is interesting to note that if a sequence of 4 bits is available, then only 16 levels can be permitted; 5 bits provide 32 levels, 6 bits 64 levels, and so on. The more levels that are defined the less is the quantization noise, or approximation, and the greater is the bandwidth needed to transmit the extra pulses. I shall return to this subject in Chap. 4. Suffice it to state here that the signal-to-quantization-noise power ratio increases exponentially with bandwidth. This is an extremely efficient exchange of bandwidth for signal-to-noise ratio, and approaches the theoretical maximum attain-able.
Tuller4 has demonstrated that only coded systems, such as PCM, can efficiently exchange bandwidth for signal-to-noise ratio. The uncoded systems such as frequency modulation (FM) or PPM exhibit an improvement in signal-to-noise power ratio
If 4 pulses per sample are to be transmitted, as described above, then a transmission bandwidth of 4/m Hz is required. For binary PCM, the signal-to-noise power ratio must be sufficient to distinguish between the presence or absence of a pulse; this requires y'S+ NIVN= 2.5
The same quantity of information (81m bit/s) may be transmitted using a bandwidth equal to 21m Hz. This is achieved by transmitting 2 quaternary pulses per sample. reduce the transmission bandwidth by increasing the average signal power and using a multilevel code. Clearly this is the reverse process of quantizing, and it is necessary therefore to increase the power exponentially for a linear decrease in the transmission bandwidth.
1.6 1.4 1.2 l!l 1.0 '0 0.8
>
0.6 0.4 0.2
Communication by Sampling 39
° --- --
tt t
Sampling instantst t t t
PPM
PWM
Binary PCM
1°.1.1.111.0.0.111.0.0.11°.1.1.11°.1.0.°1°.0.1.°1
0.7 0.9 0.9 0.7 0.4 0.2 Volts
Fig. 3·3 The various pulse modulation systems.
noise within the transmission system and on the practical difficulties of designing the coding/decoding equipments.
In conclusion to this section we refer the reader to Fig. 3-3, which illustrates the types of pulse modulation systems that have been described. In the remaining sections of this chapter we shall use PAM as the basis of discussion, since it is the easiest scheme to visualize. However, the points raised are equally applicable to the other pulse modulation systems.
3.3 SAMPLING SPECTRA
It is necessary, for a full explanation of the sampling process, to provide a theoretical basis. This we do here. The mathematical equations that describe the frequency
40 Pulse-Code Modulation
-Fig. 3-4 The spectrum of the sampling signal: (a) sampling waveform; (b) spectrum of sampling waveform: Sn = Ad sinc (nd); (c) the sinc function.
The sampling waveform S(t) has previously been considered as a series of impulses with an infinitesimal width, as shown in Fig. 3-1. This is referred to as ideal, or instantaneous sampling, and is physically impossible. It is more reasonable to assume that S(t) is a periodic series of pulses of fixed amplitude, finite width T, and period T seconds, as shown in Fig. 3-4a. This is known as nonideal, or natural sampling.
The sampling waveform S(t) is defined by
Communication by Sampling 41
when -T12
<
t< TI2 when TI2<
t< T- TI2The frequency spectrum of this waveform is not continuous: it exists only at certain discrete values of w = 21T'1T. Thus, the spectrum of S(t) may be graphically repre-sented as a series of equally spaced lines, whose heights are proportional to the ampli-tudes of the discrete frequency components, as shown in Fig. 3-4h. It is instructive The spectrum represented by Eg. (3.4) can indeed only exist at certain discrete values, and has an envelope specified by the function in parentheses, as shown in Fig.
3-4b. This is the sinc function, as identified below, and in Fig. 3-4c. It occurs throughout the mathematics of sampling, and is well known to communications engi-neers and physicists6 alike. If x
=
nd, then sinc x is given by. () sin1T'x
smc x = - - - (3.5)
1T'X
The Fourier transform of a periodic function may be taken in the limit to be equivalent to the Fourier transforms of the individual components. It is possible to express a
42 Pulse-Code Modulation
d='?':'=.!.
T 5
r=iJ T= ~
r=~ T=l
____ ,yT,mm OliIT
Fig.3-5 The frequency spectrum of the sampling waveform S(t).
But
Hence
Wo=-27T T
s(t)
= L
Snejnwotn= -00
Taking the Fourier transform on both sides gives:
Ys(t)
=
FL
Snejnwotn=-OCI
It can be shown that7 the Fourier transform of ejnwo t is given by Y'(:ejnwot) = 27T 6(w - nwo)
Substituting this result into Eq. (3.6):
Ys(t) = 27T
L
Sn 6(w - nwo)n= -00
Substituting Eq. (3.4) into Eq. (3.7):
or ) d ~ sin n7Td ~( ) Ys(t =27TA ~ d u w-nwo
n=-oo n7T
(3.6)
(3.7)
(3.8) This equation represents the frequency spectrum of the waveform 5tt), and consists of impulses that exis~ at w = 0, ±wo, ±2wo, . . . ,±nwo. The amplitude of the impulse located at w
=
nwo is given by 27TAd sinc (nd). This spectrum is illustrated for d =Ys,
and d = ~o, in Fig. 3-5.We return to Eq. (3.2) in order to obtain an expression for the frequency spectrum of the sampled signalfs(t). Using the convolution theorem it is possible to state that the multiplication of two functions in the time domain is equivalent to the tion of their spectra in the frequency domain. Thus Eq. (3.2) rewritten as the convolu-tion of the two frequency spectra becomes:
7 This is proved in ,B. P. Lathi, Communication Systems, Wiley, New York, 1968.
Communication by Sampling 43
Fig.3·6 Waveforms and spectra for natural sampling: (a) message signal; (b) sampling signal;
(c) pulse amplitude modulated signal.
44 Pulse-Code Modulation
sideband situated on either side of the impulses defined by Eq. (3.8). These impulses, it will be remembered, exist at 0, ±wo, ±2wo, ±3wo, . .. ,±nwo, and their amplitude varies as sinc (nd). The complete information about the message signal is contained within each sideband, and is retrievable by filtering.