{"id":12590,"date":"2025-12-27T19:42:52","date_gmt":"2025-12-28T00:42:52","guid":{"rendered":"https:\/\/jeremyclark.ca\/wp\/?p=12590"},"modified":"2025-12-28T07:10:18","modified_gmt":"2025-12-28T12:10:18","slug":"carrier-synchronization-with-costas-squaring-loops-simulation","status":"publish","type":"post","link":"https:\/\/jeremyclark.ca\/wp\/telecom\/carrier-synchronization-with-costas-squaring-loops-simulation\/","title":{"rendered":"Carrier Synchronization with Costas & Squaring Loops -Simulation"},"content":{"rendered":"\n

Introduction<\/h2>\n\n\n\n

Costas & Squaring Loops are key components in reception of Telecommunication Signals. They are used to synchronize a local generated carrier with an incoming received carrier. We can simulate and study their operation using Scicos (Ref.1<\/a>). First we generate a BPSK waveform at R=100bit\/sec to use with our two Demodulators (Ref.2<\/a>).<\/p>\n\n\n\n

BPSK Waveform Generation<\/h2>\n\n\n\n
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FIg.1 Scicos Schematic for BPSK Generation PN Data Source R=100bit\/sec Fc=1KHz<\/figcaption><\/figure>\n\n\n\n
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Fig.2 A=BPSK PN Data 2^5-1=31bit R=100bit\/sec<\/figcaption><\/figure>\n\n\n\n
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Fig.3 Spectrum of BPSK PN Data Fc=1KHz +\/- 100Hz<\/figcaption><\/figure>\n\n\n\n

Figure 1 shows a Scicos Schematic that can be used to generate BPSK at R=100bit\/sec with carrier Fc=1KHz. This is identical to the schematic used in Ref.2<\/a> for QPSK by just removing the Demux & Q branch. Figure 2 shows the PN data waveform repeating every 31bits=31*0.01sec=310msec. Figure 3 shows the output spectrum centered at Fc=1KHz +\/-100Hz to first nulls. Notice the absence of a discrete line at Fc that a receiver could lock on!<\/p>\n\n\n\n

Costas Loop Scicos Simulation<\/h2>\n\n\n\n
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Fig.4 Scicos Schematic for Costas Loop Demodulation BPSK PN Data A<\/figcaption><\/figure>\n\n\n\n
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Fig.5 Costas Loop Demodulated PN Data<\/figcaption><\/figure>\n\n\n\n
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Fig.6 Close Up Costas Demodulated PN Data<\/figcaption><\/figure>\n\n\n\n

Figure 4 shows a Scicos Schematic for a Costas Loop BPSK demodulator. Following the derivation in Ref.3<\/a>, assume that the VCO is almost locked to the incoming carrier with a small phase error of e. On the I branch:
m(t)cos(wct)*cos(wct+e)=0.5m(t)[cos(e)+cos(2wct+e)]
After the LPF (removes 2wct component):
m(t)cos(wct)*cos(wct+e)=0.5*m(t), (cos(e)=1 for small e)
On the Q branch:
m(t)cos(wct)*sin(wct)=0.5m(t)[sin(e)+sin(2wct+e)]
After the LPF (removes the 2wct component):
m(t)cos(wct)*sin(wct+e)=0.5m(t)sin(e)
On the VCO branch:
0.5m(t)cos(e)*0.5m(t)sin(e)=0.25(m(t)^2)sin(2e)
The VCO LPF picks out the DC component to drive the VCO.
Figures 5 & 6 show the demodulated BPSK data waveform identical to what was sent. <\/p>\n\n\n\n

Squaring Loop Scicos Simulation<\/h2>\n\n\n\n
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Fig.7 Squaring Loop for Carrier Recovery & Data Demodulation<\/figcaption><\/figure>\n\n\n\n
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Fig.8 Squaring Loop Demodulation PN Data<\/figcaption><\/figure>\n\n\n\n

Figure 7 shows the Squaring Loop used for carrier recovery and data demodulation. The input signal is first squared:
[m(t)*coswct]^2=0.5*([m(t)]^2 + cos(2wct))
Thus this gives a low frequency component plus a perfect carrier at twice wc. This is then Band Pass Filtered to give just the carrier component at 2wc. Finally this is divided by 2 (not so easy!!!!) to yield a perfect version of the received carrier. Recall that the transmit spectrum had no discrete line at Fc! This recovered carrier now multiplies the input received signal and Low Pass Filtered to remove m(t)\/2 as in the Costas Loop I Branch. Figure 8 shows the data recovery identical to what was transmitted.<\/p>\n\n\n\n

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