Chapter 5 Digital Bandpass Modulation and Demodulation Techniques
Chapter 5 Digital Bandpass Modulation and Demodulation Techniques Binary Amplitude Shift Keying Pages 212-219 219
The analytical signal for binary amplitude shift keying (BASK) is: s BASK (t (t)) = s baseband (t) ) sin (2π f C t) (S&M Eq. 5.1) The signal s baseband (t) ) can be any two shapes over a bit time T b but it is usually a rectangular signal of amplitude 0 for a binary 0 and amplitude A for binary 1. Then BASK is also known as on-off off keying (OOK). MS Figure 3.5 T b 0 1 1 0 1
The binary amplitude shift keying (BASK) signal can be simulated in Simulink. s BASK (t) ) = s baseband (t) ) sin 2π2 f C t (S&M Eq. 5.1) Sinusoidal carrier f C = 20 khz, A c = 5 V Multiplier BASK signal baseband binary PAM signal 0,1 1 V, r b = 1 kb/sec
A BASK signal is a baseband binary PAM signal multiplied by a carrier (S&M Figure 5-3). 5 Unmodulated sinusoidal carrier Baseband binary PAM signal BASK signal
The unipolar binary PAM signal can be decomposed into a polar PAM signal and DC level (S&M Figure 5-4). 5 Unipolar binary PAM signal 0 1 V Polar binary PAM signal ± 0.5 V DC level 0.5 V
The spectrum of the BASK signal is (S&M Eq. 5.2): S BASK (f S BASK (f (f)) = F( s ASK (t) ) ) = F( s baseband (t) ) sin (2π f C t) ) (f)) = 1/2 j (S baseband (f f C ) + S baseband (f + f C ) ) The analytical signal for the baseband binary PAM signal is: s baseband (t S baseband (f (t)) = s PAM (t) ) + A/2 (S&M Eq. 5.3) (f)) = S PAM (f) ) + A/2 δ(f) (S&M Eq. 5.4) Therefore by substitution (S&M Eq. 5.5): S BASK (f (f)) = 1/ 2j ( S PAM (f f C ) + A/2 δ(f f C ) S PAM (f + f C ) A/2 δ(f + f C ) )
The bi-sided power spectral density PSD of the BASK signal is (S&M Eq. 5.7): G BASK (f (f)) = 1/4 G PAM (f f C ) + 1/4 G PAM (f + f C ) + A 2 /16 δ(f f C ) + A 2 /16 δ(f + f C ) For a rectangular polar PAM signal (±( A): G PAM (f (f)) = (A/2) 2 / r b sinc 2 (π f / r b ) (S&M Eq. 5.8) MS Figure 3.7
The single-sided sided power spectral density PSD of the BASK signal is: G PAM (f G BASK (f (f)) = (A/2) 2 / r b sinc 2 (π f / r b ) (f)) = 1/2 G PAM (f + f C ) + A 2 /8 δ(f + f C ) Carrier 20 khz MS Figure 3.7 1 khz r b = 1 khz sinc 2
The bandwidth of a BASK signal as a percentage of total power is double that for the same bit rate r b = 1/T b binary rectangular PAM (MS Table 2.1 p. 22) (MS Table 3.1 p. 91). Bandwidth (Hz) Percentage of Total Power 2/T b 90% 3/T b 93% 4/T b 95% 6/T b 96.5% 8/T b 97.5% 10/T b 98%
Chapter 5 Digital Bandpass Modulation and Demodulation Techniques Binary Phase Shift Keying Pages 219-225 225
The analytical signal for binary phase shift keying (BPSK) is: s BPSK (t) = s baseband (t ) = s baseband sin (2π f C t + θ) ) (S&M Eq. 5.11) (t)) = + A b i = 1 s baseband (t) ) = A b i = 0 0 MS Figure 3.13 +180 T b 0 0 +180 0 0 1 1 0
The BPSK signal initial phase θ = 0, 0, +A is a phase shift = 0 and A A is a phase shift = +180 s BPSK (t) = s baseband (t ) = s baseband sin (2π f C t) (S&M Eq. 5.11) (t)) = + A b i = 1 s baseband (t) ) = A b i = 0 0 MS Figure 3.13 +180 T b 0 0 +180 0 0 1 1 0
The binary phase shift keying (BPSK) signal can be simulated in Simulink. PM modulator BPSK signal Fig312.mdl baseband binary PAM signal 0,1 V, r b = 1 kb/sec
The Phase Modulator block is in the Modulation, Communication Blockset but as an analog passband modulator not a digital baseband modulator.
The Phase Modulator block has the parameters of a carrier frequency f C in Hz, initial phase in radians and the phase deviation constant in radians per volt (rad( / V). f C = 20 khz initial phase φ o = π phase deviation k p = π / V
The Random Integer Generator outputs 0,1 V and with a initial phase = π and a phase deviation constant = π/v, the phase output φ of the BPSK signal is: b i = 0 φ = π + 0(π/V) = π b i = 1 φ = π + 1(π/V) = 2π 2 = 0
The spectrum of the BPSK signal is (S&M Eq. 5.13): S BPSK (f S BPSK (f (f)) = F( s PSK (t) ) ) = F(s baseband (t) ) sin 2π2 f C t) (f)) = 1/2 j (S baseband (f f c ) + S baseband (f + f C ) ) The analytical signal for the baseband binary PAM signal is: s baseband (t S baseband (f (t)) = s PAM (t) (S&M Eq. 5.12) (f)) = S PAM (f) Note that there is no DC level in s PAM (t) ) and therefore by substitution: S BPSK (f (f)) = 1/ 2j ( S PAM (f f C ) S PAM (f + f C ) )
The bi-sided power spectral density PSD of the BPSK signal is (S&M Eq. 5.13) G BPSK (f) = 1/4 G PAM (f f C ) + 1/4 G PAM (f + f C ) For a rectangular polar PAM signal (±( A): G PAM (f (f)) = A 2 / r b sinc 2 (π f / r b ) (S&M Eq. 5.8 modified) MS Figure 3.14
The single-sided sided power spectral density PSD of the BPSK signal is: G BPSK (f) = 1/2 G PAM (f + f C ) (f)) = A 2 / r b sinc 2 (π f / r b ) G PAM (f No carrier MS Figure 3.14 r b = 1 khz sinc 2
The bandwidth of a BPSK signal as a percentage of total power is double that for the same bit rate r b = 1/T b binary rectangular PAM (MS Table 2.1 p. 22) and the same as BASK (MS Table 3.1 p. 91) (MS Table e 3.5 p. 100) Bandwidth (Hz) Percentage of Total Power 2/T b 90% 3/T b 93% 4/T b 95% 6/T b 96.5% 8/T b 97.5% 10/T b 98%
Chapter 5 Digital Bandpass Modulation and Demodulation Techniques Binary Frequency Shift Keying Pages 219-225 225
The analytical signal for binary frequency shift keying (BFSK) is: s BFSK (t s BFSK (t f c f (t)) = A sin (2π (f C + f) ) t + θ) ) if b i = 1 (t)) = A sin (2π (f C f) ) t + θ) ) if b i = 0 f c + f f c + f f c f T b MS Figure 3.9 f c f 0 1 1 0 0
The BFSK signal initial phase θ = 0 0 s BFSK (t s BFSK (t (t)) = A sin (2π (f C + f) ) t) if b i = 1 (t)) = A sin (2π (f C f) ) t) if b i = 0 f c f f c + f f c + f f c f T b MS Figure 3.9 f c f 0 1 1 0 0
The binary frequency shift keying (BFSK) signal can be simulated in Simulink. BFSK signal FM Modulator Fig38.mdl baseband binary PAM signal 0,1 V, r b = 1 kb/sec
The Frequency Modulator block is in the Modulation, Communication Blockset but as an analog passband modulator not a digital baseband modulator.
The Frequency Modulator block has the parameters of a carrier frequency f C in Hz, initial phase in radians and the frequency deviation constant in Hertz per volt (Hz/V). f C = 20 khz initial phase = 0 frequency deviation = 2000
The Random Integer Generator outputs 0,1 V but is offset to ±1 and with a initial phase = 0 and a frequency deviation constant = 2000 Hz/V, the frequency shift f of the BFSK signal is: b i = 0 d i = 1 f = 0 1(2000 Hz/V) = 2000 Hz b i = 1 d i = +1 f = 0 + 1(2000 Hz/V) = +2000 Hz
The BFSK signal can be decomposed as (S&M Eq. 5.14): s BFSK (t (t)) = s baseband1 (t) sin (2π (f C + f) ) t + θ) ) + baseband2 (t) sin (2π (f C f) ) t + θ) s baseband2 1 0 0 1 1 f c + f f c f f c f f c + f f c + f
The BFSK signal is the sum of two BASK signals: s BFSK (t (t)) = s baseband1 (t) sin (2π (f C + f) ) t + θ) ) + baseband2 (t) sin (2π (f C f) ) t + θ) s baseband2 From the linearity property,, the resulting single-sided sided PSD of the BFSK signal G BFSK (f) ) is the sum of two G BASK (f) PSDs with f = f C ± f: G BFSK (f (f)) = (A/2) 2 / 2 r b sinc 2 (π f / r b ) + A 2 /8 δ(f) f c f f c + f
The single-sided sided power spectral density PSD of the BFSK signal is: G BFSK (f (f)) = (A/2) 2 / 2r2 b sinc 2 (π (f C + f) / r b ) + A 2 /8 δ(f C + f) + (A/2) 2 / 2r2 b sinc 2 (π (f C f)/ r b ) + A 2 /8 δ(f C f) carriers MS Figure 3.10 f = 2 khz r b = 1 khz sinc 2
Minimum frequency shift keying (MFSK) for BFSK occurs when f = 1/2T b = r b /2 Hz. G BFSK (f (f)) = (A/2) 2 / 2r2 b sinc 2 (π (f C + f) / r b ) + A 2 /8 δ(f C + f) + (A/2) 2 / 2r2 b sinc 2 (π (f C f)/ r b ) + A 2 /8 δ(f C f) carriers MS Figure 3.11 f = 500 Hz r b = 1 khz sinc 2
This BFSK carrier frequency separation 2 f = 1/T b = r b Hz is the minimum possible because each carrier spectral impulse is at the null of the PSD of the other decomposed BASK signal and thus is called minimum frequency shift keying (MFSK). carriers MS Figure 3.11 2 f = 1000 Hz r b = 1 khz sinc 2
The bandwidth of a BFSK signal as a percentage of total power is greater than that of either BASK or BPSK by 2 f2 Hz for the same bit rate r b = 1/T b (MS Table 3.3 p. 95). For MFSK 2 f 2 = r b = 1/T b Hz. Bandwidth (Hz) Percentage of Total Power 2 f + 2/T b 90% 2 f + 3/T b 93% 2 f + 4/T b 95% 2 f + 6/T b 96.5% 2 f + 8/T b 97.5% 2 f + 10/T b 98%
Chapter 5 Digital Bandpass Modulation and Demodulation Techniques Coherent Demodulation of Bandpass Signals Pages 225-236 236
The development of the optimum receiver for bandpass signals utilizes the same concepts as that for the optimum baseband receiver: Optimum Filter h (t) = k s(it t) o b Correlation Receiver
The optimum filter H o (f) ) and the correlation receiver are equivalent here also, with s 1 (t) = s(t) ) for symmetrical signals and r(t) ) = γ s(t) ) + n(t) ) where γ is the communication channel attenuation and n(t) ) is AWGN. The energy per bit E b and the probability of bit error P b is (S&M p. 226): E b it b (i-1)t b itb 2 2 = γ s(t) γ s(t) dt = γ s(t) dt P b 2 E b = Q No (i-1)t b
The matched filter or correlation receiver is a coherent demodulation process for bandpass signals because not only is bit time (T( b ) as for baseband signals required but carrier synchronization is also needed. Carrier synchronization requires an estimate of the transmitted frequency (f C ) and the arrival phase at the receiver (θ):( s(t)=sin(2π 1 f C t + θ)
BPSK signals are symmetrical with: s 1T (t) = s 2T (t) = A sin(2π f c t) S&M Eqs.. 5.15-5.19 5.19 E E it [ f ] = ± A γ sin (2π t) dt b, BPSK C (i-1) T itb 2 2 2 2 γ A γ A T = [ 1 cos (4π f t) ] dt = 2 2 b, BPSK C (i-1) T P b, BPSK b b b 2 2 2 E b γ A Tb = Q = Q No No 2 b
For this analysis of E b, PSK for BPSK signals it is assumed that the transmitter produces an integer number of cycles within one bit period T b : S&M Eq. 5.17 2 2 itb γ A E = b, BPSK [ 1 cos (4π fc t) ] dt 2 0 E (i-1) T b 2 2 2 2 b γ A Tb γ A = cos (4π f t) dt 2 2 b, BPSK C (i-1) T it b
However, even for a non-integer number of cycles within one bit period T b if 1 / f C = T C << T b : S&M Eq. 5.17 it b (i-1) T E b cos (4π f t) dt << T C 2 2 2 2 b γ A Tb γ A = cos (4π f t) dt 2 2 b, BPSK C (i-1) T b it b insignificant
BPSK signals are symmetrical with: s 1T (t) = s 2T (t) = A sin(2π f c t) S&M Eqs.. 5.15-5.19 5.19 and s 1 (t) = sin (2π f c t) it b a(i T )= γ s(t) s(t) dt i b i 1 (i-1) T b a(i T = 2 b)+a(i T ) τ 1 b opt = 0 2 S&M Eq. 4.67 S&M Eq. 4.71
BPSK signals are symmetrical with: s 1T (t) = s 2T (t) = A sin(2π f c t) S&M Eqs.. 5.15-5.19 5.19 τ = 0 opt S&M Figure 4-164
For this analysis of E d, ASK for BASK signals it is assumed that the transmitter produces an integer number of cycles within one bit period: S&M Eq. 5.27 2 2 itb γ A E = d, BASK [ 1 cos (4π fc t) ] dt 2 0 E (i-1) T b 2 2 2 2 b γ A Tb γ A = cos (4π f t) dt 2 2 d, BASK C (i-1) T it b
BASK OOK signals are not symmetrical with: s 1T (t) = A sin(2π f c t) s 2T (t) = 0 S&M Eqs.. 5.22, 5.23 and s 1 (t) = sin(2π f c t) ) s 2 (t) = 0 it b [ ] a(i T )= γ s (t) s (t) s (t) dt i b i 1 2 (i-1) T b itb 2 γ A Tb 1 Tb 1 2 2 (i-1) T a(i )= γ A s (t) dt = a (i T ) = 0 b S&M Eq. 5.24-5.26 5.26 b a(i T )+a(i T ) γ A T 2 4 2 b 1 b b τ opt = =
BASK signals in general may not be symmetrical with: s 1T (t) = A 1 sin(2π f c t) s 2T (t) = A 2 sin(2π f c t) and s 1 (t) s 2 (t) = sin (2π f c t) where the amplitude is arbitrary. it b [ ] a(i T )= γ s (t) s (t) s (t) dt i b i 1 2 (i-1) T b itb 2 γ Ai i Tb i 1 2 (i-1) T a(i )= γ A s (t) dt = b τ opt T b S&M Eq. 4.67 S&M Eq. 4.71 ( ) a(i 2 T )+a(i γ A +A 1 T ) = = 2 4 b b 1 2 T b
BFSK signals are not symmetrical with: s T (t) ) = A sin(2π (f c ± f) t) S&M Eq. 5.30 it b b [ ] E = A γ (sin (2π f + f) t sin (2π f f) t )) dt d, BFSK C C (i-1)t 2 2 E = γ A T if f + f = n / T and f f = n / T d, BFSK b C 1 b C 2 b P b, BFSK 2 2 E d, FSK γ A T = b = Q Q 2 No 2 No 2 S&M Eq. 5.31 S&M Eq. 5.32
For this analysis of E d, FSK for BFSK signals it is assumed that the transmitter produces an integer number of cycles within one bit period: S&M Eq. 5.31 2 2 it 0 b 2 2 γ A E d, BFSK = γ A Tb cos (4π ( fc + f) t) dt 2 (i-1) Tb 0 2 2 itb γ A cos (4π ( fc f) t) dt 2 itb 0 2 2 γ A sin (2π ( f + f) t) sin (2π ( f f) t) dt (i-1) T b (i-1) T b C C
BFSK signals are not symmetrical with: s T (t) ) = A sin(2π (f c ± f) t) S&M Eq. 5.30 and s 1 (t) s 2 (t) = sin(2π (f c + f) ) t) sin(2π (f c f) ) t) it b [ ] a(i T )= γ s (t) s (t) s (t) dt i b i 1 2 (i-1) T b itb 2 γ Ai i Tb i i 2 (i-1) T a(i )= γ A s (t) dt = b T b S&M Eq. 5.31 S&M Eq. 5.33 ( A ) a(i 2 T γ A b)+a(i 1 T ) 1 2 T b τ opt = = b = 0 2 4 if A = A 1 2
A comparison of coherent BPSK, BFSK and BASK illustrates the functional differences, but BFSK and BASK uses E d and not E b : P b, BPSK 2 2 2 E b γ A Tb = Q = Q No No E b, BPSK = γ A 2 2 2 T b P b, BFSK 2 2 E d, FSK γ A Tb = Q = Q 2 No 2 No E = γ A T 2 2 d, BFSK b P b, BASK 2 2 E d, ASK γ A Tb = Q = Q 2 No 4 No E d, BASK = γ A 2 2 2 T b
The normalized E b, FSK = E b, PSK = γ 2 A 2 T b / 2 (S&M Eq. 5.24) and E b, ASK = γ 2 A 2 T b / 4 (S&M Eq. 5.36) so that: P b, BPSK 2 2 2 E b, PSK γ A Tb = Q = Q No No E b, BPSK = 2 2 γ A T 2 b P b, BFSK 2 2 E b, FSK γ A Tb = Q = Q No 2 No E b, BFSK = 2 2 γ A T 2 b P b, BASK 2 2 E b, ASK γ A Tb = Q = Q No 4 No 2 2 γ A T 4 Thus there are no practical advantages for either coherent BFSK or BASK and BPSK is preferred (S&M p. 236). E b, BASK = b
For the same P b BPSK uses the least amount of energy, BFSK requires twice as much and BASK four times as much energy: 2 2 2 E b, PSK γ A Tb Pb, BPSK = Q = Q No No Argument of 2 2 E b, FSK γ A Tb P = Q = Q Q should be b, BFSK No 2 No as large as possible to 2 2 E γ A b, ASK T minimize P b b Pb, BASK = Q = Q No 4 No
Chapter 3 Bandpass Modulation and Demodulation Optimum Bandpass Receiver: The Correlation Receiver Pages 81-85 85
The matched filter or correlation receiver for bandpass symmetrical signals can be simulated in Simulink: MS Figure 3.1
The matched filter or correlation receiver for bandpass asymmetrical signals can also be simulated in Simulink: MS Figure 3.2
The alternate but universal structure which can be used for both asymmetric or symmetric binary bandpass signals can be simulated in Simulink: MS Figure 3.3
Chapter 3 Bandpass Modulation and Demodulation Binary Amplitude Shift Keying Pages 86-92
Binary ASK (OOK) coherent digital communication system with BER analysis: Threshold MS Figure 3.4
EE4512 Analog and Digital Communications Chapter 4 The BER and P b comparison (MS Table 3.2, p. 91): Table 3.2 Observed BER and Theoretical P b as a Function of E d / N o in a Binary ASK Digital Communication System with Optimum Receiver E d / N o db BER P b 0 0 12 2.9 10-3 2.53 10-3 10 1.12 10-2 1.25 10-2 8 3.46 10-2 3.75 10-2 6 7.65 10-2 7.93 10-2 4 1.335 10-1 1.318 10 2 1.863 10-1 1.872 10 0 2.387 10-1 2.394 10 10-1 10-1 10-1
Chapter 3 Bandpass Modulation and Demodulation Binary Phase Shift Keying Pages 98-103
Binary PSK coherent digital communication system with BER analysis: MS Figure 3.12
EE4512 Analog and Digital Communications Chapter 4 The BER and P b comparison (SVU Table 3.5, p. 167): Table 3.5 Observed BER and Theoretical P b as a Function of E b / N o in a Binary PSK Digital Communication System with Optimum Receiver E b / N o db BER P b 0 0 10 0 4.05 10 8 1 10-4 2.06 10 6 2.5 10-4 2.41 10 4 1.31 10-2 1.25 10 2 3.35 10-2 3.75 10 0 8.19 10-2 7.93 10 10-6 10-4 10-3 10-2 10-2 10-2
Chapter 3 Bandpass Modulation and Demodulation Binary Frequency Shift Keying Pages 92-98 98
Binary FSK coherent digital communication system with BER analysis: f C f f C + f MS Figure 3.9
The BER and P b comparison (MS Table 3.4, p. 98): Table 3.4 Observed BER and Theoretical P b as a Function of E d / N o in a Binary FSK (MFSK) Digital Communication System with Optimum Receiver E d /N o db BER P b 0 0 12 2.5 10-3 2.5 10-3 10 1.29 10-2 1.25 10-2 8 3.50 10-2 3.75 10-2 6 8.04 10-2 7.93 10-2 4 1.352 10-1 1.314 10 2 1.833 10-1 1.872 10 0 2.456 10-1 2.393 10 10-1 10-1 10-1
The BER and P b performance comparison for BASK, BPSK and BFSK (MFSK): E d / N o db BER P b 10 1.12 10-2 1.25 10-2 BASK 8 3.46 10-2 3.75 10-2 E b / N o db BER P b 10 0 4.05 10-6 BPSK 8 1 10-4 2.06 10-4 E d / N o db BER P b 10 1.29 10-2 1.25 10-2 BFSK 8 3.50 10-2 3.75 10-2
BER and P b comparison using E b with E b, ASK = γ 2 A 2 T 2 b / 4 and thus reduced by 10 log (0.5) 33 db or: E b / N o db BER P b 7 1.12 10-2 1.25 10-2 BASK 5 3.46 10-2 3.75 10-2 E b / N o db BER P b 10 0 4.05 10-6 BPSK 8 1 10-4 2.06 10-4 E b / N o db BER P b 10 1.29 10-2 1.25 10-2 BFSK 8 3.50 10-2 3.75 10-2 BASK performs better than BFSK but BPSK is the best.
Chapter 5 Digital Bandpass Modulation and Demodulation Techniques Differential (Noncoherent) Phase Shift Keying Pages 267-271 271
Differential (noncoherent) phase shift keying (DPSK) is demodulated by using the received signal to derive the reference signal. The DPSK protocol is: Binary 1: Transmit the carrier signal with the same phase as used for the previous bit. Binary 0: Transmit the carrier signal with its phase shifted by 180 relative to the previous bit.
The one-bit delayed reference signal r i-1 (t) is derived from the received signal r i (t) ) and if the carrier frequency f C is an integral multiple of the bit rate r b : r (t) = γ A sin (2π f (t T ) + θ) i 1 r (t) = γ A sin (2π f t+ θ) i 1 C C The output of the integrator for a binary 0 and binary 1 then is z(it b ) = ± γ 2 A 2 T b / 2 (S&M Eqs.. 5.91 and 5.93) b S&M Eqs. 5.88 and 5.89
DPSK signals have an equivalent bit interval T DPSK = 2 T b. The probability of bit error for DPSK signal is different than that for coherent demodulation of symmetric or asymmetric signals and is: P b, DPSK 1 E 1 E DPSK b, DPSK = exp = exp 2 2 No 2 No 2 2 γ A Tb Eb, DPSK = S&M Eq. 5.102 2
Chapter 3 Bandpass Modulation and Demodulation Differential Phase Shift Keying Pages 130-135 135
Binary DPSK noncoherent digital communication system with BER analysis: one-bit continuous BPF delay MS Figure 3.33
Binary DPSK noncoherent digital communication system differential binary encoder Simulink Subsystem: XOR one bit sample delay MS Figure 3.34
Simulink Logic and Bit Operations provides the Logical Operator block:
Simulink Logical Operator blocks can be selected to provide multiple input AND, OR, NAND, NOR, XOR, NXOR, and NOT functions:
The Logic and Bit Operations can be configured as scalar Boolean binary (0, 1) or M-ary (0, 1 M 1) 1 1) vector logic functions. Here scalar Boolean binary data is used.
The XOR logic generates the DPSK source coding: Table 3.16 Input Binary Data b i, Differentially Encoded Binary Data d i, and Transmitted Phase φ i (Radians) for a DPSK Signal. b i d i-1 d i φ i one-bit startup 1 0 1 1 1 0 XOR logic 0 1 0 π 0 0 1 0 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 0 π 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0
The Signal Processing Blockset provides the Filtering, Analog Filter Design block:
The Signal Processing Blockset provides the analog bandpass filter (BPF) specified as a 9-pole 9 Butterworth filter with cutoff frequencies of 19 khz and 21 khz centered around the carrier frequency f C = 20 khz. rad/s
The Butterworth BPF is used for the noncoherent receiver. The coherent receiver uses the integrator as a virtual BPF: MS Figure 3.33 DPSK MS Figure 3.12 PSK
The BER and P b comparison (MS Table 3.17, p. 134): Table 3.17 Observed BER and Theoretical P b as a Function of E b / N o in a Binary DPSK Digital Communication System with Noncoherent Correlation Receiver Statistical variation due to small sample E b / N o db BER P b size 0 0 12 0 6.6 10-8 10 2 10-4 2.3 10-5 8 5.1 10-3 1.8 10-3 6 2.61 10-2 9.3 10-3 4 7.91 10-2 4.06 10-2 2 1.559 10-2 1.025 10-1 0 2.393 10-1 1.839 10-1
BER and P b comparison between noncoherent, source coded DPSK and coherent BPSK: E b / N o db BER P b 12 0 6.6 10-8 DPSK 10 2 10-4 2.3 10-5 8 5.1 10-3 1.8 10-3 E b / N o db BER P b 10 0 4.05 10-6 BPSK 8 2 10-4 2.06 10-4 6 2.5 10-3 2.41 10-3 BPSK performs better than DPSK but requires a coherent reference signal. DPSK performs nearly as well as BPSK at high SNR.
Chapter 5 Digital Bandpass Modulation and Demodulation Techniques M-ary Bandpass Techniques: Quaternary Phase Shift Keying Pages 274-286 286
Quaternary phase shift keying (M-ary, M = 2 n = 4 or QPSK) source codes dibits b i-1 b i as a symbol with one possible protocol as: b i-1 b i = 11 A sin(2π f C t + 45 ) b i-1 b i = 10 A sin(2π f C t + 135 ) b i-1 b i = 00 A sin(2π f C t + 225 ) b i-1 b i = 01 A sin(2π f C t + 315 ) The Gray code is used as for M-ary M PAM to improve + the BER performance by mitigating adjacent symbol error. The symbols are best displayed as a constellation plot S&M Figure 5-365 modified
Quaternary phase shift keying (QPSK) displayed as a constellation plot Note the signs on the sine reference axes. cos + Constellation Gemini constellation points sin S&M Figure 5-365 modified
Chapter 3 Bandpass Modulation and Demodulation Multilevel (M-ary) Phase Shift Keying Pages 117-123 123
The QPSK signal can be simulated in Simulink using Subsystems to simplify the design. QPSK I-Q I Q correlation receiver 4-level Gray coded bit to symbol 4-level Gray coded symbol to bit MS Figure 3.22
The random binary data source is converted to an M = 4 level Gray encoded symbol by a Simulink Subsystem. MS Figure 3.22
The M = 4 level symbol (0, 1, 2 and 3) is Gray encoded by a Lookup Table Block from the Simulink Blockset. MS Figure 3.22
The M = 4 level symbol (0, 1, 2 and 3) is Gray encoded by a Lookup Table Block from the Simulink Blockset.
The M = 4 level symbol (0, 1, 2, and 3) is Gray encoded by a Lookup Table Block by mapping [0, 1, 2, 3] to [0, 1, 3, 2] Gray coding 00 00 01 01 10 11 11 10
The M = 4 level symbol is inputted to the Phase Modulator block with a carrier frequency f C = 20 khz, initial phase φ o = π/4 and a phase deviation factor k p = π/2 / V
The M = 4 level symbol (0, 1, 2, 3) and the carrier frequency f C = 20 khz, initial phase φ o = π/4 and a phase deviation factor k p = π/2 / V produces the phase shifts: d i = 0 φ = π/4 + 0(π/2) = π/4 d i = 1 φ = π/4 + 1(π/2) = 3π/43 d i = 2 φ = π/4 + 2(π/2) = 5π/45 d i = 3 φ = π/4 + 3(π/2) = 7π/47 MS Figure 3.22
The modulation phase shifts are the phase angle φ of the sinusoidal carrier A sin (2π f C t + φ) ) in QPSK. d i = 0 φ = π/4 + 0(π/2) = π/4 45 d i = 1 φ = π/4 + 1(π/2) = 3π/43 135 d i = 2 φ = π/4 + 2(π/2) = 5π/45 225 d i = 3 φ = π/4 + 3(π/2) = 7π/47 315 MS Figure 3.22
The QPSK signal can be resolved into In-phase (I, cosine) and Quadrature (Q, sine) constellation components. For example, Plot if φ = π/4 = 45 : Quadrature s 1 (t) = A sin(2π f C t + 45 ) ) = cos A / 2/ 2 [ I sin (2π f C t) + Q cos (2π f C t) ] = A / 2/ 2 [ sin (2π f C t) + cos (2π f C t) ] + sin In-phase S&M Figure 5-365 modified
The QPSK signal is derived from Gray coded dibits with 00 00 (0), 01 01 (1), 10 11 (3) and 11 10 (2). 10 11 11 01 01 11 r b = 1 kb/sec 00 00 M = 4 Delay 3 2 0 2 0 1 1 2 ± 5 V, f C = 2 khz, r S = 500 Hz QPSK signal T S = 2 msec
The QPSK signal can be decomposed into I and Q BPSK signals which are orthogonal to each other. ± 5 V QPSK signal, f C = 2 khz, r S = 500 b/sec ± 5 / 2 2 = 3.536 V Binary PSK signal, sine carrier (I) ± 5 / 2 2 = 3.536 V Binary PSK signal, cosine carrier (Q) T S = 2 msec
The orthogonality of the I and Q components of the QPSK signal can be exploited by the universal coherent receiver. The orthogonal I and Q components actually occupy the same spectrum without interference. The coherent reference signals are: Quadrature In-phase s 1 (t) = cos (2π f C t + θ) ) s 2 (t) = sin (2π f C t + θ) S&M Figure 5-405
The orthogonality of the QPSK signals can be shown by observing the output of the quadrature correlator to the I and Q signal. its γ A z(n 1 TS)= d I sin(2π fct) + dq cos(2π fct) cos (2π fct) dt 2 (i-1) TS 0 its γ A z(n 1 TS)= d I sin(2π fct) cos (2π fct) dt + 2 (i-1) TS S&M Eq. 5.109 its γ A 2 d Q cos (2π fc t) dt 2 γ A T z(n T )= d 2 2 S 1 S Q (i-1) T S z 1 (nt S )
The probability of bit error P b and the energy per bit E b for a QPSK signal is the same as that as for a BPSK signal but with a I and Q carrier amplitude of A / 2. P P b, BPSK b, QPSK 2 2 2 E b, PSK γ A Tb = Q = Q No No 2 2 2 E b, QPSK γ A TS = Q = Q No 2 No note T S S&M Eq. 5.117 E E b, BPSK b, QPSK = = γ A 2 2 4 T 2 2 2 γ A T b S note T S z 1 (nt S )
Since T S = 2 T b BPSK and QPSK have the same P b but QPSK can have twice the data rate r b = 2 r S within the same bandwidth because of the orthogonal I and Q components. P 2 2 2 E b γ A Tb = P = Q = Q No No b, BPSK b, QPSK E = E = b, BPSK b, QPSK γ A 2 2 2 T b z 1 (nt S )
QPSK coherent digital communication system with BER analysis: 4-Level Gray coded bit to symbol MS Figure 3.22 4-Level Gray coded symbol to bit
QPSK coherent digital communication system uses a 4-level Gray coded bit to symbol converter Simulink Subsystem. MS Figure 2.43
QPSK coherent receiver uses an I-Q I Q correlator Simulink Subsystem MS Figure 3.24
The I-Q I Q correlation receiver is the universal structure with an integration time equal to the symbol time T S. MS Figure 3.24 correlation receiver
The output of the I-Q I Q correlation receiver is a dibit and converted to an M = 2 n = 4 level symbols (0, 1, 2, and 3). dibits M-ary M scaling MS Figure 3.24 correlation receiver
The BER and P b comparison for 4-PSK 4 (QPSK): Table 3.11 Observed BER and Theoretical Upper-Bound of P b as a Function of E b / N o in a Gray coded 4-PSK 4 (QPSK) Digital Communication System with Optimum Receiver E d /N o db BER P b 0 0 12 0 10 10 0 10 8 2 10-4 10 6 2.3 10-3 2.4 10-3 4 1.20 10-2 1.25 10 2 3.62 10-2 3.75 10 0 7.65 10-2 7.85 10 10-2 10-2 10-2
The single-sided sided power spectral density PSD of the QPSK signal uses r s = r b /2 and is: G QPSK (f G PAM (f (f)) = 1/2 G PAM (f - f C ) (f)) = A 2 / r s sinc 2 (π f / r s ) No carrier r s = 500 s/sec, r b = 1 kb/sec Sinc 2 MS Figure 3.25
The single-sided sided power spectral density PSD of BPSK has double the bandwidth than that for QPSK for the same bit rate r b = 1/T b, No carrier r b = 1 khz Sinc 2 MS Figure 3.14
The bandwidth of a QPSK signal as a percentage of total power is half that for the same bit rate r b = 1/T b BPSK signal since r s = r b /2 or T s = 2T2 b (MS Table 3.9). Bandwidth (Hz) Percentage of Total Power 2/T s 1/T b 90% 3/T s 1.5/T b 93% 4/T s 2/T b 95% 6/T s 3/T b 96.5% 8/T s 4/T b 97.5% 10/T s 5/T b 98%
Chapter 5 Digital Bandpass Modulation and Demodulation Techniques M-ary Bandpass Techniques: 8-Phase Shift Keying Pages 286-292 292
M-ary phase shift keying (M = 8 or 8PSK) source codes tribits b i-2 b i-1 b i as a symbol with one possible protocol as: b i-2 b i-1 b i = 000 A sin(2π f C t + 0 ) 0 Constellation Plot b i-2 b i-1 b i = 001 A sin(2π f C t + 45 ) b i-2 b i-1 b i = 011 A sin(2π f C t + 90 ) b i-2 b i-1 b i = 010 A sin(2π f C t + 135 ) b i-2 b i-1 b i = 110 A sin(2π f C t + 180 ) b i-2 b i-1 b i = 111 A sin(2π f C t + 225 ) b i-2 b i-1 b i = 101 A sin(2π f C t + 270 ) b i-2 b i-1 b i = 100 A sin(2π f C t + 315 ) θ I, Q = 0, ± 1/ 2, ± 1 s(t) ) = A [ I sin (2π f C t) + Q cos (2π f C t) ] S&M Figure 5-435 Quadrature cos sin In-phase
The correlation receiver for 8-PSK 8 uses four reference signals: (t)) = sin (2π f C t + n 45 + 22.5 ) φ s ref n (t n = 0, 1, 2, 3 φ = 22.5,, 67.5, 112.5,, 157.5 S&M Eq. 5.124
The output from any one of the four correlators is: 1 it z(n T )= γ A sin (2π f t + θ) sin (2π f t + φ)dt (i-1) TS 0 its its γ A z(n 1 TS)= cos (θ φ) dt cos (4π fct + θ + φ) dt 2 γ A TS z(n 1 TS )= cos (θ φ) 2 S S C C (i-1) T S S&M Eq. 5.125 (i-1) T S z 1( n T S)
The correlator output is > 0 if θ φ < 90 and < 0 if not because of the cos (θ( φ) ) term. For example, if s 6 (t) is received, the ABCD correlator sign output is: +. The patterns of signs are unique and can be decoded to b i-2 b i-1 b i (S&M Tables 5-75 7 and 5-8) 5 D: s ref 4 (t) C: s ref 3 (t) A: s ref 1 (t) B: s ref 2 (t) z 1( n T S)
The probability of symbol error P S for coherently demodulated M-ary M PSK is: 2 A T S 2 π PS coherent M-ary PSK 2Q sin M 4 No M E 2 b π P 2Q 2 log S coherent M-ary PSK 2 M sin M 4 N M o P s S&M Eq. 5.126 S&M Figure 5-465 E b / N o db
The probability of symbol error P S for coherently demodulated M-ary M PSK is: S&M Figure 5-465 P s E b /N o db
The probability of symbol error P S must be related to probability of bit error P b for consistency. If Gray coding is used, assume that errors will only be due to adjacent symbols.. Thus each symbol error produces only one bit in error and log 2 (M 1) correct bits or: 1 Pb errors due to adjacent symbols = PS S&M log M Eq. 5.127 However for M-ary M PSK with M > 4 the assumption of errors being due to only adjacent symbols is invalid. For the worst case there are M 1 incorrect symbols and in M / 2 of these a bit will different from the correct bit so that: 2 1 M PS P b PS S&M Eq. 5.129 log M 2 (M 1) 2
Chapter 5 Digital Bandpass Modulation and Demodulation Techniques M-ary Bandpass Techniques: Quaternary Frequency Shift Keying Pages 292-298 298
The analytical signal for quaternary (M-ary, M = 2 n = 4) frequency shift keying (QFSK or 4-FSK) 4 is: FSK(t) = A sin (2π (f C + 3 f) 3 ) t + θ) ) if b i-1 b i = 11 FSK(t) = A sin (2π (f C + f) ) t + θ) ) if b i-1 b i = 10 FSK(t) = A sin (2π (f C f) ) t + θ) ) if b i-1 b i = 00 FSK(t) = A sin (2π (f C 3 f) ) t + θ) ) if b i-1 b i = 01 s 4-FSK s 4-FSK s 4-FSK s 4-FSK MS Figure 3.19 f C + 3 f3 f C f f C 3 f f C + f f C + 3 f3 11 00 01 10 11
Chose f C and f so that if there are a whole number of half cycles of a sinusoid within a symbol time T S for M = 4 for orthogonality of the signals so that a correlation receiver can be utilized. FSK(t) = A sin (2π (f C + 3 f) 3 ) t + θ) ) if b i-1 b i = 11 FSK(t) = A sin (2π (f C + f) ) t + θ) ) if b i-1 b i = 10 FSK(t) = A sin (2π (f C f) ) t + θ) ) if b i-1 b i = 00 FSK(t) = A sin (2π (f C 3 f) ) t + θ) ) if b i-1 b i = 01 s 4-FSK s 4-FSK s 4-FSK s 4-FSK MS Figure 3.19
The correlation receiver for 4-FSK 4 uses four reference signals: (t)) = sin (2π (f C + n f) t) s ref n (t n = ±1, ±3 S&M Figure 5-495
The probability of symbol error P S for coherently demodulated M-ary M FSK is: 2 A T s P S coherent M-ary FSK (M 1)Q M 4 2 No P s P = S coherent M-ary FSK E b (M 1) Q log 2 M M 4 N o S&M Figure 5-515 51 E b /N o db S&M Eq. 5.132
The probability of symbol error P S for coherently demodulated M-ary M FSK is: S&M Figure 5-515 51 P s E b /N o db
Chapter 3 Bandpass Modulation and Demodulation Multilevel (M-ary) Frequency Shift Keying Pages 110-116 116
4-FSK coherent digital communication system with BER analysis: MS Figure 3.18
The dibits are converted to a symbol and scaled. The data is not Gray encoded. For M-ary M FSK symbol errors are equally likely among the M 1 correlators and there is no advantage to Gray encoding. MS Figure 3.18
4-FSK coherent digital communication system with BER analysis: 4-FSK correlation receiver MS Figure 3.18
4-FSK coherent digital communication system with BER analysis: 4-FSK correlation receiver MS Figure 3.20
The 4-FSK 4 correlation receiver has four correlators with an integration time equal to the symbol time T S.
The symbols are converted to dibits. The original data is not Gray encoded and is therefore not Gray decoded. MS Figure 3.18
The BER and P b comparison for 4-FSK: 4 Table 3.10 Observed BER and Theoretical Upper Bound of P b as a Function of E b / N o in 4-level 4 FSK Digital Communication System with Optimum Receiver E d /N o db BER P b 0 0 12 0 10 10 0 10 8 1 10-4 10 6 5.1 10-3 4.8 10-3 4 2.26 10-2 2.52 10-2 2 5.97 10-2 7.54 10-2 0 1.209 10-1 1.586 10 10-1
The single-sided sided power spectral density PSD with a minimum carrier frequency deviation (MFSK) for M-ary M FSK is f = 1/2T S = r S /2. For MFSK the carriers should be spaced at multiples of 2 f 2 f = 1/T S = r S (S&M Eq. 5.131 is incorrect). Here f = 2 r S = 1 khz M = 4 f = 1 khz r s = 500 s/sec, r b = 1 kb/sec Sinc 2 MS Figure 3.21
The bandwidth of a M-ary M FSK signal as a percentage of total power (MS Table 3.9). Bandwidth (Hz) Percentage of Total Power 2( M 1) f + 4/T s 95% 2 (M 1) f + 6/T s 96.5% 2 (M 1) f + 8/T s 97.5% 2 (M 1) f + 10/T s 98% For MFSK: f = 1/2T S = r S /2 M = 2 n and r S = r b /n
Chapter 5 Digital Bandpass Modulation and Demodulation Techniques M-ary Bandpass Techniques: Quadrature Amplitude Modulation Pages 298-301
The analytical signal for quadrature amplitude modulation (QAM) has I-Q I Q components: s QAM (t (t)) = I sin (2π f C t) ) + Q cos (2π f C t) A QAM signal has both amplitude and phase components which can be shown in the constellation plot. 16-ary QAM Q S&M Figure 5-535 53 I
An M-ary M PSK signal also has I-Q I Q components but the amplitude is constant and only the phase varies: s QAM (t (t)) = I sin (2π f C t) ) + Q cos (2π f C t) 16-ary PSK S&M Figure 5-545 54 16-ary QAM Q Q I I
The orthogonality of the I and Q components of the QAM signal can be exploited by the universal coherent receiver. The orthogonal I and Q components actually occupy the same spectrum without interference. The coherent reference signals are: Quadrature In-phase s 1 (t) = cos (2π f C t) ) s 2 (t) = sin (2π f C t) S&M Figure 5-555 55
An upper-bound for the probability of symbol error P S for coherently demodulated M-ary M QAM is: P S coherent M-ary QAM 4 3 E s Q (M 1) No S&M Eq. 5.135 QAM BER curve M = 256 M = 4
An M-ary M QAM constellation plot shows the stability of the signaling and the transition from one signal to another: 256-ary QAM 16-ary QAM
Chapter 3 Bandpass Modulation and Demodulation Quadrature Amplitude Modulation Pages 123-130 130
QAM coherent digital communication system with BER analysis: 4 bit to I,Q symbol 16-QAM correlation QAM receiver MS Figure 3.26 16-level symbol to bit
QAM coherent digital communication system with BER analysis: 4 bit to I-Q I Q symbol Simulink S subsystem. MS Figure 3.27
QAM coherent digital communication system with BER analysis: Table 3.12 I and Q output amplitudes Input I Q Input I Q Input I Q 0 1 1 5 1 3 10 3 1 1 3 1 6 3 1 11 3 3 2 1 3 7 3 3 12 1 1 3 3 3 8 1 1 13 3 1 4 1 1 9 1 3 14 1 3 15 3 2 I LUT ± 1 to ± 3 Q LUT symbol 0 to 15 MS Figure 3.27
QAM coherent digital communication system with BER analysis: QAM modulator Simulink S subsystem. MS Figure 3.27
QAM coherent digital communication system with BER analysis: 16-QAM correlation receiver Simulink S subsystem. MS Figure 3.30
QAM coherent digital communication system with BER analysis: Table 3.14 I, Q Symbol LUT 16-level output amplitudes I Q Output I Q Output 1 1 11 3 1 1 1 2 9 3 2 0 1 3 14 3 3 4 1 4 15 3 4 6 2 1 10 4 1 3 2 2 8 4 2 2 2 3 12 4 3 5 2 4 13 4 4 7
QAM coherent digital communication system with BER analysis: 16 level symbol to 4 bit Simulink S subsystem. MS Figure 3.31
QAM coherent digital communication system with BER analysis: 16 level symbol to 4 bit Simulink S subsystem. MS Figure 3.31
The single-sided sided power spectral density PSD of the 16-ary QAM has a bandwidth of 1/M that of a PSK signal with the same data rate r b. r s = 250 s/sec, r b = 1 kb/sec Sinc 2 no discrete component at f C = 20 khz MS Figure 3.32
1 khz BPSK PSD r b = 1 kb/sec 16-ary QAM PSD 250 Hz r b = 1 kb/sec M = 4 r S = 250 s/sec
The bandwidth of an M-ary M QAM signal as a percentage of total power is 1/n that for the same bit rate r b = 1/T b BPSK signal since r s = r b /n or T s = ntn b where M = 2 n (MS Table 3.14). Bandwidth (Hz) Percentage of Total Power 2/T s 2/nT b 90% 3/T s 3/nT b 93% 4/T s 4/nT b 95% 6/T s 6/nT b 96.5% 8/T s 8/nT b 97.5% 10/T s 10/nT b 98%
16-QAM coherent digital communication system received I-Q Q components can be displayed on as a signal trajectory or constellation plot. real-imaginary (a + b j) conversion to complex polar (M exp(jθ)) conversion Figure 3.42
The Real-Imaginary to Complex conversion block is in the Math Operations, Simulink Blockset Figure 3.42
The constellation plot (scatter plot) and signal trajectory are Comm Sinks blocks from the Communications Blockset Figure 3.42
The 16-ary QAM I-Q I Q component constellation plot with E b /N o (MS Figures 3.43, 3.45). signal transitions signal points decision boundaries
The 16-ary QAM I-Q I Q component constellation plot with E b /N o = 12 db, P b 10-4 (MS Figures 3.44, 3.46 (top))
The 16-ary QAM I-Q I Q component constellation plot with E b /N o = 6 db, P b = 3.67 x 10-4 (MS Figures 3.44, 3.46 (bot( bot))
End of Chapter 5 Digital Bandpass Modulation and Demodulation Techniques