Chapter 6: Introduction to Digital Communication

Transcription

1 93 Chapter 6: Introduction to Digital Communication 6.1 Introduction In the context o this course, digital communications include systems where relatively high-requency analog carriers are modulated y relatively low-requency digital inormation signals (digital modulation) and systems involving the transmission o digital pulses (digital transmission). Digital transmission systems transport inormation in digital orm thereore they require a physical acility etween the transmitter and receiver such as a metallic wire pair, a coaxial cale or an optical ier cale. In digital modulation systems, the carrier acility could e a physical cale or it could e ree space. In this chapter, the student will e irst introduced to inormation theory parameters ollowed y the introduction o several orms o digital modulation system. The inal part o the chapter is where the student will learn more on digital transmission systems. 6.2 Inormation Theory Parameters Inormation Capacity, Bits and Bit Rate Inormation theory is a study o the eicient use o andwidth to propagate inormation through electronic communication systems. Inormation theory can e used to determine the inormation capacity o a data communication system. Inormation capacity is a measure o how much inormation can e propagated through a communication system and it is a unction o andwidth and transmission time. I.e. inormation capacity represents the numer o independent symols that can e carried through a system in a given unit o time. The most asic digital symol used to represent inormation is the inary digit, or it. Bit rate is simply the numer o its transmitted during one second and is expressed in its per second (ps). In 1928, Hartley s Law is developed to show the relation etween inormation capacity, andwidth and transmission time. Hartley Law: I B t (6.1) Where I = inormation capacity (ps) B = andwidth (Hz) t = transmission time (seconds)

2 94 Then in 1948, mathematician Claude E. Shannon relates the inormation capacity o a communication channel to andwidth and signal-to-noise ratio. The higher the signal-tonoise ratio, the higher the inormation capacity is. I.e. etter perormance is produced. Shannon limit or inormation capacity: S S I = B log or I = 3.32B log (6.2) N N Where I = inormation capacity (ps) B = andwidth (Hz) S = signal-to-noise ratio (unitless) N M-ary Encoding M-ary is a term derived rom word inary. M simply represents a digit that corresponds to the numer o conditions, levels or cominations possile or a given numer o inary variales. For example, a digital signal with our possile conditions (voltage levels, requencies, phases and so on) is an M-ary system where M = 4. I there are eight possile conditions, M = 8 and so orth. The numer o its necessary to produce a given numer o conditions is expressed mathematically as N = log 2 M (6.3) Where N = numer o its necessary M = numer o conditions or levels possile with N its Equation (6.3) can e rearranged to express the numer o conditions possile with N its 2 N = M (6.4) For example, with one it, only 2 conditions are possile. With two its, 4 conditions are possile, with three its, 8 conditions are possile, and so on Baud Baud is a term oten misunderstood and commonly conused with it rate. Bit rate reers to the rate o change o a digital inormation signal, which is usually inary. Baud is also a rate o change; however,

3 95 Baud reers to the rate o change o a signal on the transmission medium ater encoding and modulation have occurred. I.e. aud is a unit o transmission rate, modulation rate or symol rate and thereore the terms symols per second and aud are oten used interchangealy. Mathematically, aud is expressed as aud 1 = (6.5) t s Where aud = symol rate (symol per second) t = time o one signaling element or symol (seconds) s Comparison etween aud and it rate can e urther explained as the ollowing. Binary signals are generally encoded and transmitted one it at a time in the orm o discrete voltage levels representing logic 1 (high) or 0 (low). A aud is also transmitted one at a time; however, a aud may represent more than one inormation it. I.e. the aud o a data communication system may e consideraly less than the it rate. In inary encoding systems, aud and it rate (ps) are equal. In higher-level encoding systems, it rate is always greater than aud. Worked Example Assume we wanted to transmit the decimal numer 201. This can e represented in inary as Using inary (2-level) encoding system, these its are transmitted serially as a sequence o equal-time-interval pulses that are either 1 or 0. I each it interval is 1 ms, then the it rate is 1000 ps (1/1ms). The aud rate is also 1000 ps or 1000 aud (1000 symols per second). Now, let a 4-level encoding system represents 2 its o data as dierent voltage levels. Since there are 4 possile cominations o 2 its, we will have 4 dierent voltage levels. For example, With this system, would e divided into groups o 11/00/10/01. Thereore, the transmitted signal would e voltage levels o 3, 0, 2 and 1 respectively. I each voltage level occurs at 1 ms interval, the aud rate is still 1000 aud ecause there is only one symol per time interval. (I.e symols per second)

4 96 The it rate now is 2000 ps since each symol represents 2 its (1000 x 2). I.e. we have douled the it rate while keeping the aud rate constant. In addition, the transmission time is also shortened. It takes 8 ms to transmit 8-it inary word using inary system, ut it only takes 4 ms to transmit the word using 4-level encoding system Minimum Bandwidth According to H. Nyquist, inary digital signals can e propagated through an ideal noiseless transmission medium at a rate equal to two times the andwidth o the medium. The minimum theoretical andwidth necessary to propagate a signal is called the minimum Nyquist andwidth or sometimes the minimum Nyquist requency. Mathematical representation: = 2B (6.6) Where = Bit rate / Channel capacity (ps) B = minimum Nyquist andwidth (Hz) The relationship etween andwidth and it rate also applies to the opposite situation. For a given andwidth (B), the highest theoretical it rate is 2B. However, i more than two levels are used or signaling, more than one it may e transmitted at a time, and it is possile to propagate a it rate that exceeds 2B. Using multi-level signaling, equation (6.6) ecomes 2 = 2B log M (6.7) Where = Bit rate / Channel capacity (ps) B = minimum Nyquist andwidth (Hz) M = numer o conditions or level Worked example Using previous worked example parameters, the minimum andwidth required to transmit the signal on inary encoding system can e calculated as = 2 B B = / 2 = 1000 / 2 = 500 Hz For the 4-level encoding system, the minimum andwidth is similar. = 2B log B = 2log = = 2log 4 2(2) 2 M = 2 M Hz I.e. or a same andwidth, we can propagate a higher it rate using multi-level system.

5 97 Equation (6.7) can e urther simpliied to solve or the minimum andwidth necessary to pass M-ary digitally modulated carrier: Sustituting Equation (6.3) into Equation (6.8) B = (6.8) log 2 M B = (6.9) N Where N = numer o its encoded into each signaling element In addition to that, since aud is the encoded rate o change, it also equals the it rate divided y the numer o its encoded into one signaling element. Thereore aud = (6.10) N I.e. the aud and the ideal minimum Nyquist andwidth have the same value and are equal to the it rate divided y the numer o its encoded. This is true or all orms o digital modulation except requency-shit keying. 6.3 Digital Modulation Digital modulation is the transmittal o digitally modulated analog signals (carriers) etween two or more points in a communication system. Digital modulation is sometimes called digital radio ecause digitally modulated signals can e propagated through Earth s atmosphere and used in wireless communication systems. Figure 6.1 shows a simpliied lock diagram or a digital modulation system. Encoder perorms level conversion and then encodes the incoming data into groups o its that modulate an analog carrier inside modulator. The modulated carrier is iltered, ampliied and then transmitted through transmission medium to the receiver. The transmission medium can e metallic cale, optical ier cale, Earth s atmosphere or comination o two or more types o transmission systems. The received signal is iltered, ampliied and then applied to the demodulator and decoder circuits, which extracts the original source inormation rom the modulated carrier. The clock and carrier recovery circuits recover the analog carrier and digital timing (clock) signals rom the incoming modulated wave or demodulation purpose.

6 98 Figure 6.1: Simpliied lock diagram o digital modulation system In general, there are three asic digital modulation techniques, namely: Amplitude Shit keying (ASK), Frequency Shit Keying (FSK) and Phase Shit Keying (PSK). Figure 6.2 shows the output waveorm or these three digital modulation techniques. Figure 6.2: ASK, FSK and PSK modulation scheme

7 99 In ASK, the modulator puts out a urst o carrier or every logic 1, and no signal or every logic 0. In FSK, or logic 1 a higher requency carrier urst is transmitted and or logic 0 a lower requency carrier urst is transmitted, or vice versa. In PSK, logic 1 is transmitted as a urst o carrier with zero initial phase while logic 0 is transmitted as a urst o carrier with initial phase Amplitude Shit Keying (ASK) The simplest digital modulation technique is ASK, where a inary inormation signal directly modulates the amplitude o an analog carrier. ASK can e represented mathematically as A v( ask) ( t) = [1 + vm ( t)] cos( ω ct) 2 (6.11) Where v ask ( ) = ASK wave, v m (t) = digital inormation (modulating) signal ( ) t A 2 ω c = unmodulated carrier amplitude, = analog carrier radian requency The modulating signal in Equation (6.11) is a normalized inary waveorm, where +1 = logic 1 and -1 = logic 0. For logic 1 input, A vm ( t) = + 1 v( ask ) ( t) = [1 + 1] cos( ω ct) = Acos( ωct) 2 For logic 0 input, A vm ( t) = 1 v( ask ) ( t) = [1 1] cos( ω ct) = 0 2 I.e., the ASK signal is either Acos( ω ct) (ON) or 0 (OFF), which is why ASK is also called on-o keying (OOK). Figure 6.3 shows an example o ASK waveorm. Figure 6.3: ASK waveorm

8 100 For every change in the input inary data, there is one change in the ASK waveorm and the time o one it (t ) equals the time o one analog signaling element (t s ). Since the it time is the reciprocal o the it rate and the time o one signaling element is the reciprocal o the aud, thereore, the it rate in ASK modulation technique is equal to the aud. With ASK, the it rate is also equal to the minimum Nyquist andwidth B (y setting N = 1 into Equation (6.9) and Equation (6.10)) B = = = N 1 and aud = = = N Frequency Shit Keying (FSK) FSK is a orm o constant-amplitude angle modulation similar to standard requency modulation (FM) except that the modulating signal is a inary signal that varies etween two discrete voltage levels rather than a continuously changing analog waveorm. FSK is also known as inary FSK (BFSK). Mathematical expression or FSK Where ( ) = FSK wave v( sk ) t [ 2 [ + v ( t ] t] v( sk ) ( t) = c cos π c m ) (6.12) v m (t) = inary input (modulating) signal c c = peak analog carrier amplitude = analog carrier centre requency = peak change (shit) in analog carrier requency The modulating signal in Equation (6.12) is also a normalized inary waveorm, where +1 = logic 1 and -1 = logic 0. For logic 1 input, v For logic 0 input, v t) = + 1 v ( t) = cos π m ( ( sk ) t) = 1 v ( t) = cos π m ( ( sk ) c c [ 2 ( + ) t] c [ 2 ( ) t] With inary FSK, the carrier center requency is shited up and down in the requency domain y the inary input signal and the direction o the shit is determined y the polarity as shown in Figure 6.4. c

9 101 Logic 1 Logic s c m Figure 6.4: FSK in the requency domain As the inary input changes rom logic 0 to logic 1 and vice versa, the output requency shits etween two requencies: a mark or logic 1 requency ( m ) and a space or logic 0 requency ( s ). The mark and space requencies are separated rom the carrier centre requency y the peak requency deviation. Frequency deviation can e expressed mathematically as Figure 6.5 shows an example o FSK waveorm in time domain. m = s (6.13) 2 Figure 6.5: FSK waveorm Based on Figure 6.5, the time o one it (t ) is the same as the time o an FSK signaling element (t s ). I.e. the FSK it rate is equal to the aud o FSK. Again y setting N = 1 in Equation (6.10), aud = = = N 1

10 102 FSK is the exception to the rule or digital modulation, as the minimum Nyquist andwidth B is not determined using Equation (6.9). The minimum Nyquist andwidth or FSK is given as B = ( ) ( ) = + 2 s m s m Since s = 2 as in Equation (6.13), m B = ( + ) (6.14) Phase Shit Keying (PSK) PSK is another orm o angle-modulated, constant-amplitude digital modulation. PSK is an M-ary digital modulation scheme similar to conventional phase modulation except with PSK the input is a inary digital signal and there are limited numers o output phase possile. The numer o output phases is deined y M as descried in Equation (6.4) and determined y the numer o its N. The simplest orm o PSK is inary PSK (BPSK), where N = 1 and M = 2. Thereore, with BPSK, two phases are possile or the carrier. One phase represents logic 1 and the other phase represents logic 0. As the input digital / inary signal changes state, the phase o the output carrier shits etween two angles that are separated y Figure 6.6 shows a simpliied lock diagram o BPSK transmitter Binary data (modulating) Level converter (unipolar to ipolar) Balanced modulator Band pass ilter Modulated PSK output Buer sin (ω c t) Reerence carrier oscillator Figure 6.6: BPSK transmitter I +1 is assigned to input logic 1 and -1 is assigned to input logic 0, the carrier sin( ω t) is multiplied y either +1 or -1. c

11 103 For logic 1, the output BPSK signal is v ( t) = sin( ω ) while or logic 0, the ( psk ) ct ( psk ) ( t) = sin( ωct. output BPSK signal is v ) I.e. logic 1 output represents a signal that is in phase with the reerence oscillator and logic 0 output represents a signal that is out o phase with reerence oscillator. Figure 6.7 shows an example o BPSK waveorm. Figure 6.7: BPSK waveorm As inary input shits etween logic 1 and logic 0 and vice versa, the phase o the BPSK waveorm shits etween 0 0 to 180 0, respectively. For simplicity, only one cycle o the analog carrier in shown in each signaling element, although there may e anywhere etween a raction o a cycle to several thousand cycle, depending on the relationship etween the input it rate and the analog carrier requency. Note that the time o one BPSK signaling element (t s ) is equal to the time o one input it (t ), which indicates that the it rate equals the aud. As in ASK, the minimum Nyquist andwidth B or FSK is given as 2 = 2 B = 6.4 Digital Transmission Digital transmission is the transmittal o digital signals etween two or more points in a communication system. The signals can e inary or any other orm o discrete-level digital pulses. The original source inormation may e in digital orm or it could e analog signals that have een converted to digital pulses prior to transmission. With digital transmission systems, a physical acility, such as a pair or wires, coaxial cale or an optical ier cale, is required to interconnect the various points within the system. Note that digital pulses cannot e propagated through a wireless transmission system, such as Earth s atmosphere or ree space.

12 Pulse Modulation Pulse modulation is a process o sampling analog inormation signals and then converting those samples into discrete pulses and transporting the pulses rom a source to a destination over a physical transmission medium. The our predominant methods o pulse modulation are Pulse Width Modulation (PWM), Pulse Position Modulation (PPM), Pulse Amplitude Modulation (PAM) and Pulse Code Modulation (PCM). In PWM, the width o constant-amplitude pulse is varied proportional to the amplitude o the analog signal at the time the signal is sampled. In PPM, the position o a constant-width and constant-amplitude pulse is varied according to the amplitude o the sample o the analog signal. In PAM, the amplitude o a constant-width pulse is varied proportional to the amplitude o the sample o the analog signal. In PCM, the analog signal is sampled and then converted to a serial n-it inary code or transmission. Each code has the same numer o its and requires the same length o time or transmission. Figure 6.8 shows examples o PWM, PPM and PAM waveorms. Figure 6.8: PWM, PPM and PAM waveorms For PWM, the maximum analog signal amplitude produces the widest pulse and the minimum analog signal amplitude produces the narrowest pulse. For PPM, the higher the sample s amplitude, the arther to the right the pulse is positioned within the prescried time slot. The highest amplitude sample produces a pulse to the ar right while the lowest amplitude sample produces a pulse to the ar let. For PAM, the amplitude o a pulse coincides with the amplitude o the analog signal.

13 105 PAM is used as an intermediate orm o modulation with PSK and PCM, although it is seldom used y itsel. PWM and PPM are used in special-purpose communication systems mainly or the military ut are seldom used or commercial digital transmission systems. PCM is y ar the most prevalent orm o pulse modulation and will e discussed in more detail in susequent section o this chapter Pulse Code Modulation (PCM) Figure 6.9 shows simpliied lock diagram o a single-channel, simplex PCM system. PAM signal Parallel data Analog input signal Band pass ilter Sample and hold Analog to digital converter Parallel to serial converter Sample pulse Conversion clock Line speed clock Serial PCM code Regenerative repeater Serial PCM code Regenerative repeater Serial PCM code Parallel data PAM signal Serial to parallel converter Digital to analog converter Hold circuit Low pass ilter Analog output signal Line speed clock Conversion clock Figure 6.9: PCM system Band pass ilter limits the requency o analog signal to standard voice-and requency range. Sample and hold circuit samples the analog signal and converts those samples to a multilevel PAM signal. Analog-to-digital converter converts multilevel PAM samples to parallel PCM codes. Parallel-to-serial converter converts parallel PCM codes to serial inary data. Repeaters are placed at prescried distances to regenerate the data.

14 106 Serial-to-parallel converter converts serial inary data to parallel PCM codes. Digital-to-analog converter converts parallel PCM codes to multilevel PAM signals. Hold circuit and low pass ilter converts PAM signals ack to its original orm Sampling and sampling rate The unction o a sampling circuit in a PCM transmitter is to periodically sample the continually changing analog input voltage and convert those samples to a series o pulses that can more easily e converted to inary PCM code. The Nyquist sampling theorem estalishes the minimum sampling rate that can e used or a given PCM system. The theorem states that, The original analog input signal can e reconstructed at the receiver with minimal distortion i the sampling rate in the pulse modulation system is equal to or greater than twice the maximum analog input requency. Mathematical representation: (6.15) s 2 m(max) Where s = sampling rate / sampling requency m(max) = maximum analog input requency I.e. the minimum sampling rate is equal to twice the highest analog input requency Quantization Quantization is a process o converting an ininite numer o possiilities to a inite numer o conditions. In relations to this chapter, once the analog signal is sampled, quantization is a process o assigning those samples to pre-determined discrete quantization levels. The numer o quantization levels L depends on the numer o its per sample used to code the analog signal. n L = 2 (6.16) The magnitude dierence etween adjacent levels is called the quantization interval or quantum or resolution. The resolution is equal to the voltage o the minimum step size, which in turn is equal to the voltage o the least signiicant it o the PCM code. It can e represented mathematically as

15 107 max min = (6.17) L 1 Where = resolution, = maximum analog input signal max min = minimum analog input signal In most cases, the likelihood o a sample voltage is exactly the same as one o the quantization level values is remote. Thereore, each sample voltage is rounded o (quantized) to the closest availale level. This process leads to an error called quantization error or quantization noise. It is the distortion introduced during quantization process when the analog sample voltage is not exact value o the quantized level. Mathematical representation: = [ x( t)] [ q( t)] (6.18) Where Q e Q e = quantization error / quantization noise [ x ( t)] = magnitude o analog sample voltage [ q ( t)] = magnitude o the closest quantized level Maximum quantization error is given y Q e (max) = ± (6.19) 2 Quantization error can e reduced y increasing the numer o quantization levels, ut this will increase the andwidth required to transmit the signal. Signal-to-quantization noise ratio (SQR): SQR = (6.20) Q e In deciel, 2 v v SQR = 10log = log 2 (6.21) q /12 q Where v = rms signal voltage q = quantization interval Encoding This is the process where each quantized sample is digitally encoded into n-its codes, where n maye any positive integer greater than 1. n = log 2 L (6.22)

16 108 The codes currently used or PCM are sign-magnitude codes, where the most signiicant it (MSB) is the sign it and the remaining its are used or magnitude. Tale 6.1 shows an n-it PCM code where n equals 3. Tale 6.1: 3-it PCM code Sign it Magnitude / alue it MSB is used to represent the sign o sample where logic 1 represent positive value sample while logic 0 represent negative value sample. Figure 6.10 shows all parameters related to 3-it PCM system. Figure 6.10: Quantization level, resolution, quantization error, sign it, and magnitude it

17 109 Figure 6.11 shows an analog waveorm, sampling pulse, the corresponding sampled signal (PAM), quantized signal and PCM code or each sample. Figure 6.11: (a) Input signal, () sampling pulse, (c) PAM signal, (d) quantized signal and (e) PCM code

18 Dynamic Range The numer o PCM its transmitted per sample is determined y several variales, which includes maximum allowale input amplitude, resolution and dynamic range. Dynamic range (DR) is the ratio o the largest possile magnitude to the smallest possile magnitude that can e decoded y the digital-to-analog converter (DAC) in the receiver. Mathematical representation: DR = max max = min (6.23) Where max = maximum voltage that can e decoded y DAC = = resolution min The relationship etween DR and the numer o its in a PCM code: n 2 1 DR (6.24) For a minimum numer o its, n 2 1 = DR (6.25) Where n = numer o its in a PCM code, excluding sign it Rearranging Equation (6.25), we can solve or n y taking logs: n log 2 = log( DR + 1) nlog 2 = log( DR + 1) log( DR +1) n = log 2 DR can also e expressed in deciels: max n DR ( d) = 20log = 20log( 2 1) (6.26) min Coding Eiciency Coding eiciency is a numerical indication o how eiciently a PCM code is utilized. It is a ratio o the minimum numer o its required to achieve a certain dynamic range to the actual numer o PCM its used. I.e.: Min numer o its ( including sign it) Coding eiciency = 100 (6.27) Actual numer o its ( including sign it)

19 Companding Companding is a process o compressing and then expanding. With companded systems, the higher-amplitude analog signals are compressed (ampliied less than the loweramplitude signals) prior to transmission and then expanded (ampliied more than the lower-amplitude signals) in the receiver. Companding is a means o improving the dynamic range o a communication system. There are two methods o analog companding or PCM system: µ-law companding and A-Law companding µ-law companding Used in the US and Japan. The compression characteristics or µ-law: Where out = compressed output amplitude max max ln( 1+ µ in / max ) out = (6.28) ln(1 + µ ) = maximum uncompressed analog input amplitude in = amplitude o the input signal at a particular instant o time µ = parameter used to deine the amount o compression Figure 6.12 shows the compression curve or several values o µ. Note that the higher the µ, the more compression. Figure 6.12: µ-law compression characteristics

20 A-Law companding Used in Europe and other parts o the world. The compression characteristics or A-Law: out out Ain / max = max or in 1 0 (6.28a) 1+ ln A A 1+ ln( Ain / max ) 1 in = max or 1 (6.28) 1+ ln A A max max Figure 6.13 shows the compression curves or several values o A. Figure 6.13: A-Law compression characteristics For an intended dynamic range, A-Law companding has a slightly latter SQR than µ- Law. However, A-Law companding is inerior to µ-law in terms o small-signal quality Delta Modulation PCM With conventional PCM, each code is a inary representation o oth sign and the magnitude o a particular sample. Thereore, multiple-it codes are required to represent the many values that the sample can e. With delta modulation, rather than transmit a coded representation o the sample, only a single it is transmitted, which simply indicates whether that sample is larger o smaller than the previous sample.

21 113 I the current sample is smaller than the previous sample, logic 0 is transmitted. I the current sample is larger than the previous sample, logic 1 is transmitted Delta Modulation Transmitter Figure 6.14 shows a lock diagram o a delta modulation transmitter. Figure 6.14: Delta modulation transmitter The input analog is sampled and converted to a PAM signal, which is compared to the output o the DAC. The output o the DAC is a voltage equal to the regenerated magnitude o the previous sample (stored in the up-dowm counter as a inary numer) The up-down counter is incremented or decremented depending on whether the previous sample is larger or smaller than the current sample. It is clocked at a rate equal to the sample rate (i.e. updated ater each comparison) Figure 6.15 shows the ideal operation o a delta modulation encoder. Figure 6.15: Ideal operation o a delta modulation encoder

22 114 Initial conditon: Up-down counter is zeroed, DAC ouput = 0. When the irst sample is taken and converted to PAM signal, it is compared to zero volts. The output o the comparator is logic 1 (current sample is larger in amplitude than the previous sample). Next clock pulse, the counter is updated (incremented to a count o 1). The DAC now outputs a voltage equal to the magnitude o minimum step size / resolution. The second sample is now compared to the new DAC output, and so on. Based on Figure 6.15, the up-down counter ollows the input analog sample (incremented) until the output o the DAC exceeds the analog sample amplitude; then it will egin counting down (decremented) until the output o the DAC drops elow the sample amplitude. Each time the up-down counter is incremented, logic 1 is transmitted, and each time the up-down counter is decremented, logic 0 is transmitted Delta Modulation Receiver Figure 6.16 shows the lock diagram o a delta modulation receiver. Figure 6.16: Delta modulation receiver The receiver almost identical to the transmitter except or the comparator. As the logics 1 and 0 are received, the counter is incremented or decremented accordingly. Consequently, the output o the DAC in the reciever is identical to the output o the DAC in the transmitter (Figure 6.16). With delta modulation, each sample requires the transmission o only one it, thereore the it rates associated with delta modulation are lower than conventional PCM systems. However, there are two prolems associated with delta modulation that do not occur with conventional PCM: slope overload and granular noise.

23 115 Slope overload: Figure 6.17: Slope overload distortion Occurs when the analog input signal changes at a aster rate than the DAC can maintain. The slope o the analog signal is greater than the delta modulator can maintain. Solutions: increase the clock requency or increase the magnitude o the minimum step size (resolution). Granular noise: Figure 6.18 contrasts the original and reconstructed signals associated with delta modulation system. Figure 6.18: Granular noise When the original signal has a relatively constant amplitude, the reconstructed signal has variations that were not present in the original signal. This is called granular noise. It can e reduced y decreasing the step size (resolution) Note that to reduce the granular noise, a small resolution is needed while to reduce the slope overload, a large resolution is required. I.e. a compromise is necessary. Granular noise is more prevelant in analog signals that have gradual slope and whose amplitudes vary only a small amount. Slope overload is more prevalent in analog signals that have steep slopes or whose amplitudes vary rapidly.