# MODULATION Systems (part 1)

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2 Lecture PSD [db] The NRZ PSD is shown in Figure 3. The vertical axis is normalized in terms of A 2 T. Null points corresponding to zero power appear at frequencies corresponding to various symbol rates for the case of an infinitely large bandwidth..3 Transmission path noise Noise on a transmission path or in a receiver can be broadly classified into thermal noise, generated from resistors, and artificial noise, from sources such as automobiles that reach the receiving antenna. For relatively clean links like satellite channels and cables, the noise on the transmission path can be considered to be mostly thermal. As shown in Figure 4, the temporal waveform of thermal noise is random in nature; its amplitude distribution follows a Gaussian distribution (normal distribution), whose probability distribution function (PDF) is shown in Figure 5. The average amplitude m of thermal noise is. Given a noise power of amplitude /T -3/T -2/T -/T /T 2/T 3/T 4/T Figure 3: NRZ power spectrum density frequency, P(x) is the probability that the thermal noise will take on a certain amplitude. The PSD of thermal noise is given as follows. (2) The PSD of thermal noise is thus flat over all frequencies, a condition referred to as "white noise." The above temporal and frequency characteristics are reflected by thermal noise often being called "white Gaussian noise.".4 Errors on a transmission path having noise and distortion At the receiver, transmitted digital data are judged to be or based on the signal level, which is greater or less than a threshold level, at symbol sample points. The system disregards times other than the sample points, regardless of the type of waveform. Rectangular pulses, such as the NRZ and RZ pulses described earlier, have large bandwidths (infinitely large in theory). Therefore, to make efficient use of finite frequency resources, appropriate band limitations are applied to the transmission, resulting in the smooth waveform shown in Figure 6(b). If non-linear distortion and reflection occur on the transmission path, prior and subsequent symbols with respect to the symbol in question will be affected and signal levels at sample points will vary. This distortion is called Inter Symbol Interference (ISI). If the ISI is large, the margin of the threshold level with respect to noise drops, and this lesser margin becomes a source of error. If noise is added to the signal, levels may exceed the threshold level and errors may be generated as shown in Figure 6(d). -3 Figure 4: Temporal waveform of thermal noise ( =) (a) no band limitation P(x) (b) band limit m x (c) band limit + ISI error ISI When mean =, Figure 5: Gaussian probability distribution function (d) band limit + ISI + noise error Figure 6: Transmission path having distortion and noise Broadcast Technology no.4, Spring 23 C NHK STRL

3 Transmission errors are evaluated in terms of error rate. For example, one erroneous bit in a transmission of bits of information corresponds to a bit error rate (BER) of -3. We consider errors caused by noise. If we were to superpose Gaussian noise of variance (noise power) (Figure-5) on signals of amplitudes -A and +A corresponding to codes and, the amplitude distribution at a sample point would be expressed as P(x) and P(x) as shown in Figure 7. (4) The above equation can be referred to as a "," where is the " rate." The baseband bandwidth B b is (5) P(x) Area in which code is mistaken as -A +A P(x) Area in which code is mistaken as Figure 7: Error caused by Gaussian noise Amplitude The error rate Pe is the areal percentage of that part exceeding level. It is computed as follows. Pe = (probability that code is sent) (probability that code is mistaken as ) + (probability that code is sent) (probability that code is mistaken as ) Given that the probability of sending code and that of code are both /2, we can rewrite the above as follows. In the above equation, T s is symbol period and (=/ T s ) is symbol rate. As decreases, the bandwidth of the pulse series becomes smaller, and jitter (variance at the sampling point) can easily occur. As increases, the bandwidth becomes larger. The minimum bandwidth corresponding to = is called the Nyquist bandwidth. For example, the following baseband bandwidth is needed when transmitting a -Mbps NRZ binary signal at a rate of.5. B b = ( +.5) / 2 Mbps / ( bit/symbol) = 7.5 MHz The S/N of the received pulse becomes a maximum on a linear transmission channel if the frequency response of the transmitter and receiver s behave as if they share a H(f) =.5 =.5 = Here, (3) is called the co-error function. fts.5 Roll-off Characteristic h(t).5 Condition for undistorted transmission If band limitations are applied to an NRZ rectangular pulse, ISI will occur in the waveform. Considering, however, that the absence of disturbances from other samples is desirable at a sample point, we note that ISI will not occur (ISI-free transmission) if, for any pulse, the amplitude at the sampling point is non-zero while the amplitude at all other sample points for that pulse is zero. This condition is called the Nyquist criterion. A band-limited that satisfies the Nyquist criterion and its impulse response are shown in Figure 8. Here, the frequency characteristics are given as.5 Impulse Response = = Figure 8: Roll-off characteristics and pulse that satisfies the Nyquist criterion t/ts 2 Broadcast Technology no.4, Spring 23 C NHK STRL

4 Lecture in the transmission line. It is therefore common to subject the characteristics to root allocation in the transmitter and receiver as shown in Figure 9. This is called "root." NRZ It should be emphasized that the above characteristics produce an undistorted transmission for an input signal having an ideal impulse. However, as an actual transmission pulse has a width, frequency compensation using reverse characteristics of that spectrum results in a flat spectrum, like the frequency response of an impulse. This is called aperture equalization, and it can be expressed as follows in the case of NRZ. Aperture equalization is usually done together with rolloff ing. An example of characteristics that include NRZ aperture equalization is shown in Figure. 2 x/sin x (aperture) White Gaussian Noise Figure 9: Root s in a transmission system H(f) =.5 A(f) H(f) A(f) aperture equalization.5 fts NRZ Figure : Roll-off characteristics including aperture equalization The received waveform is in a disturbed state, due to the effects of distortion and noise in the transmission path. This state can be observed in the form of eye patterns. Specifically, if the received signal is displayed on an oscilloscope overlaid by the symbol period, blank areas become visible in the center of the resulting output. These - - Figure : Examples of eye patterns ( =.5; left: no noise, right: with noise) (6) are called eye patterns because of their resemblance to eyes. The size of a single eye here can be used to visually assess the extent of waveform distortion. For example, a "closed" eye means that 's and 's cannot be distinguished and the code error is large. Note that there is one eye in a single eye pattern for a binary signal and M- eyes for an M-dimensional signal. Figure shows examples of eye patterns. 2. Digital modulation systems As in analog modulation, the function of "modulation" in digital modulation is to convert the original information signal (baseband signal) into one with a frequency convenient for transmission and to vary the amplitude, frequency, or phase of the carrier. What is different from analog modulation is that the baseband signal in question may be an analog signal or a digital signal. Varying the amplitude of the carrier (as in AM) according to the 's and 's in the baseband signal is called amplitude shift keying (ASK), varying the frequency of the carrier (as in FM) is called frequency shift keying (FSK), and varying the phase is called phase shift keying (PSK). Varying both amplitude and phase, moreover, is called quadrature amplitude modulation (QAM). We here describe several basic examples of such systems, namely, BPSK, QPSK (4PSK), and 6QAM. 2. BPSK (Binary Phase Shift Keying) The most basic form of PSK is BPSK. As shown in Figure 2, BPSK transmits a binary ( and ) digital signal by having a carrier phase of correspond to. The configuration of the BPSK modulator is shown in Figure 3. BPSK is a system that sends one bit of information per symbol. Because information is conveyed by phase, BPSK is robust against level fluctuation on the transmission path. As shown in Figure 4, the BPSK spectrum is equivalent to that of the baseband signal, although it has been shifted (minus side included) to the carrier frequency. The bandwidth of BPSK is therefore twice that of the baseband signal. When inserting a, the BPSK bandwidth for the symbol rate r (=/T) and rate is (+ )r. Demodulation by coherent detection simply reverses the process of modulation, as shown in Figure 5. The carrier circuit removes the modulated component and noise component from the received signal and recovers the carrier. The system then multiplies the received signal by this carrier to obtain the baseband signal, which is then input to a clock circuit to recover the same clock as that on the modulator side. With this clock, the system samples the baseband signal, decides whether the symbols are or, and restores the digital signal. On the other hand, demodulation by incoherent detection treats the received signal delayed by one symbol as the reference phase and multiplies the received signal by Broadcast Technology no.4, Spring 23 C NHK STRL 3

5 Baseband signal T t received signal data output Carrier t carrier clock Figure 5: BPSK demodulation by coherent detection PSK modulated wave information - Figure 2: BPSK modulated waveform baseband signal T: symbol period BPSK signal cos ( t) carrier Figure 3: BPSK modulation t received signal symbol delay clock Figure 6: BPSK demodulation by incoherent detection data output - -/2T /2T BER PSD [db] fc-4/t fc-3/t fc-2/t fc-/t fc fc+/t fc+2/t fc+3/t fc+4/t Figure 4: BPSK spectrum frequency C/N (db) Figure 7: BPSK bit error rate (coherent detection; the noise bandwidth is the Nyquist bandwidth) it to obtain the baseband signal, as shown in Figure 6. Incoherent detection is frequently used in terrestrial mobile transmissions since large fluctuations in amplitude due to fading effects make it difficult to recover the carrier. Here, however, because the signal itself, which includes noise and distortion, is used as the reference phase in incoherent detection, the bit error characteristics are worse than those of coherent detection. In addition, as the information is transmitted as a phase difference with respect to the previous symbol, differential conversion of code must be performed on the transmit side before modulation. BPSK symbol error rate assuming coherent detection of a BPSK signal to which noise has been added to the transmission path can expressed using Eq. (3), since the amplitude probability for the demodulated signal follows the Gaussian distribution shown in Figure 5. Since signal power C=A 2 /2 and noise power N= above equation can be rewritten as (7), the Because one bit corresponds to one symbol in BPSK, the above symbol error rate is the same as the bit error rate P b. This is shown in Figure 7. The CN ratio specifies the noise bandwidth, which is usually taken to be the Nyquist bandwidth. The noise bandwidth of a signal with symbol rate r (symbol/sec) is r (Hz). 2.2 QPSK (Quadrature Phase Shift Keying) The QPSK system uses carrier phases at 9-degree intervals to send two bits of information per symbol, and for this reason, it is sometimes called 4-phase PSK. It is very (8) 4 Broadcast Technology no.4, Spring 23 C NHK STRL

7 Next, with respect to a sample of symbol, r I (t) takes on the following form. for each (8) () The received signal is also detected by using the recovered carrier, and the Q-axis r Q (t) becomes as follows. The phase difference for each symbol on the I-axis is thus demodulated as information. Incoherent detection of the orthogonal axis (Q-axis) in an orthogonal modulation system such as QPSK results in the following expression for r Q (t), after shifting the phase of the signal delayed by one symbol by /2 and multiplying the result by the current signal. (9) After passing the above through an, we get Then, with respect to a sample of symbol, r Q (t) becomes. (2) for each (2) (2) After coherent detection, the I-axis and Q-axis signals, r I (t) and r Q (t), are passed through low-pass s (s) to generate baseband signals only, and the transmitted amplitude and phase, A(t) and (t), are then recovered from the post- r I (t) and r Q (t). (3) (4) In incoherent detection, the received signal delayed by one symbol is multiplied by the current signal, and the I- axis signal r I (t) becomes as follows. If we extract only the signal component here, we get (5) The phase difference for each symbol on the Q-axis is thus demodulated as information. We next consider the QPSK symbol error rate and bit error rate. For coherent detection, QPSK can be viewed as BPSK performed independently on the orthogonal I and Q signals. The QPSK symbol error rate P QPSK is expressed as follows, where the error rates of the I and Q signals are denoted by P I and P Q. (22) Since P I and P Q are each the same as the BPSK symbol error rate P BPSK, the above equation can be rewritten using Eq. (7). (23) Expressing the above in terms of the CN ratio, we get Eq. (24), considering that (from Figure 9). (24) (6) Then, after passing the above through an and extracting only the baseband component, r I (t) can be approximated as Now let us consider the QPSK bit error rate P b-qpsk. In the case of gray code assignment, the st and 2nd terms of Eq. (2) correspond to -bit error, while the 3rd term corresponds to 2-bit error. Therefore by multiplying the st, 2nd, and 3rd terms by /2, /2, and 2/2, respectively, we get the following expression for P b-qpsk. (7) 6 Broadcast Technology no.4, Spring 23 C NHK STRL

8 Lecture bit stream 2bit S/P 4 level level convert 6QAM (25) In other words, for a bit error rate equal to that of BPSK, the CN ratio of QPSK is larger by 3 db. However, compared with BPSK at the same bit rate, the symbol rate of QPSK is /2 that of BPSK, which means that transmission becomes possible at /2 the bandwidth. As a result, the signal power C required to obtain a bit error rate identical to that of BPSK is the same power as in BPSK. The bit error rate of QPSK with respect to the CN ratio is shown in Figure 22. 2bit level convert 4 level Figure 23: 6QAM modulator configuration Q I QAM -4 BER BPSK QPSK Figure 24: 6QAM constellation (compared with that of BPSK) is needed to obtain an equivalent error rate. The bit error rate of 6QAM coherent detection is C/N (db) Figure 22: QPSK bit error rate (coherent detection; the noise bandwidth is the Nyquist bandwidth) 2.3 QAM Reduction of bit rate by using source coding and band compression by using a modulation system are useful when transmitting digital signals over limited bands. In general, the case whereby a signal takes on four or more points of a constellation is called multi-dimensional modulation. A digital modulation that transmits M bits per symbol is called a 2 M -dimensional modulation (for PSK, this becomes 2 M -phase PSK). The signal obtained by performing orthogonal modulation on two multi-dimensional ASKs is called Quadrature Amplitude Modulation (QAM). The configuration of a 6QAM modulation circuit is shown in Figure 23. (26) Increasing the number of levels on each axis results in higher multi-dimensional modulations such as 64QAM and 256QAM. While frequency-usage efficiency improves with level, a higher receive power is needed to obtain a given bit error rate. The following expressions give the bit error rates for 64QAM and 256QAM. : 64QAM (27) : 256QAM (28) (Shunji Nakahara) The 6QAM modulation system can transmit four bits of information per symbol. As can be seen from the constellation in Figure 24, however, the distance between each code is small, which means that a large signal power Broadcast Technology no.4, Spring 23 C NHK STRL 7

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