# Offset in Carrier and Timing

Is my understanding about carrier and timing offset right?

Suppose my transmit frequency is 100 MHz and if my reference is 1ppm; then the transmit frequency has +/-100Hz offset.

Similarly, if my receive frequency is 100 MHz and if my reference is 2 ppm; the receive frequency has +/-200Hz offset.

So the carrier frequency offset is +/- 300 Hz, i.e 600 Hz; does this have to be recovered in receiver carrier frequency recovery?

Similarly, if the symbol rate is 100KHz with reference 1 ppm then the transmitter has +/- 0.1 Hz symbol offset and the receive symbol rate is 100kHz with reference 2 ppm it has +/- 0.2 Hz symbol offset.

So the symbol frequency offset is +/-0.3 Hz, i.e 0.6 Hz. Is this symbol recovery? What is symbol phase recovery? Is there any effect of sampling frequency offset also?

• alyssaeliyah: your edit is not really an improvement. English isn't my primary language, either, but your edit actually made Steve's pretty good English worse, and introduced counter-productive formatting. – Marcus Müller Jun 1 '20 at 20:29
• Hi, Steve, does this answer your question? dsp.stackexchange.com/questions/67170/… – Dsp guy sam Jun 2 '20 at 2:46

The calculations the OP is using is correct to predict the maximum carrier and symbol timing offset due to reference clock offsets in the transmitter and receiver.

For sampling clock offsets (both in the DAC in the transmitter and the ADC in the receiver), any offset of the sampling clock also introduces a carrier frequency offset that will be in proportion to the carrier that the analog signal is at when sampled according to $$f_a/f_s$$, where $$f_a$$ is the analog carrier frequency (of offset from $$f=0$$) and $$f_s$$ is the sampling rate. Thus a ppm change in the sampling clock will result in the same ppm change of the carrier of the sampled signal, given the actual carrier the signal is at when sampled. For example if an analog signal was translated down to a 25 MHz IF (meaning a 25 MHz carrier), and was sampled with 100 MHz sampling clock, then $$f_s=100e6$$ and $$f_a=25e6$$ in this case, and the fractional frequency of the digital IF would be $$f_a/f_s = 0.25$$ samples/cycle. If the sampling clock was actually 1% higher, or 101 MHz, the fractional frequency would 1% lower given $$f_a/f_s \approx 0.2475$$. Be very careful to reference the actual analog carrier and not the digital IF which can be different in undersampling applications. For example, if the signal was at a 125 MHz analog IF and sampled with a 100 MHz clock to produce a 25 MHz digital IF, the fractional frequency with no clock error would be $$(f_a-f_s)/f_s = 0.25$$ but if the sampling clock was 1% higher, or 101 MHz, the fractional frequency would be $$(f_a-f_s)/f_s \approx 0.2376$$ which is 5% lower! This works out to to be the clock error in ppm multiplied by the ratio of the analog IF frequency to it's equivalent frequency in the first Nyquist zone, in this case $$125/25 = 5$$.

Additional carrier frequency offset can be introduced through channel effects such as Doppler due to the motion of the transmitter and receiver that also must be taken into consideration for determining the maximum carrier and symbol timing offsets.

Once at baseband, a symbol phase offset would typically be referring to the carrier phase offset itself (rotation of the symbol) since an added phase at the carrier directly results in an added phase of the symbol. Even though it would be a referred to as a carrier phase offset, it may not have originated at the original carrier of the signal but can be introduced by other factors in the processing of the signal. If this carrier phase is changing with time, then this is a carrier frequency offset given frequency is the derivative of phase with time. Symbol timing offset at baseband in contrast would refer to the offset between the ideal sampling location for minimum error rate versus the actual sampling location. (literally a sampling time offset, or delay), and there is no rotation involved when the sampling location on the baseband signal is delayed in time. Similarly if a waveform was sampled directly at baseband, only timing offset would be introduced if there were phase or frequency offsets in the sampling clock since the IF frequency at baseband = 0 (and a frequency offset in the sampling clock would generate a varying time offset on the baseband constellation but not a rotation). Any small residual carrier offsets would be affected just like the IF frequencies above if they were then sampled with a clock that has offset errors, but this will typically be negligible given the effective IF frequency is so small.

• Thanks Dan for ur detailed explanation. 1. So for carrier freq offset I have calculated to be 600 hz I have to add ur 300 hz also. But your calculation of sampling offset is with respect to the receiver? Nothing to do with the transmitter sampling offset? 2. Symbol timing offset will also be 0.6hz+ 300hz. Again should the transmitter sampling offset is not considered?3. Carrier phase offset and timing phase offset are the same? - Steve – Steve Jun 2 '20 at 2:36
• @Steve Transmitter and receiver sampling offsets in the sampling clock frequency would both contribute. Carrier phase offset is not the same as timing phase offset-- Carrier phase offset will rotate the constellation. Timing offset will be a delay of the optimum sampling point but no rotation is involved. – Dan Boschen Jun 2 '20 at 3:00
• @Steve I corrected the ADC/DAC contributions – Dan Boschen Jun 2 '20 at 10:12
• Thanks Dan for in depth analysis.This made me to go back to the simulink and try out different parameters change and verify 1. carrier freq/phase offset= freq offset due to sampling offset in Tx & Rx( ~0 for zero IF) + freq offset caused in the RF section mixers in Tx & Rx + freq offset due to doppler 2. If i am sampling at zero IF and if i see the rotation of constellation is it because of the carrier frequency offset introduced by the analog section? 3. If i am not doing carrier recovery at all then all the carrier freq/phase offset will get added to the timing freq/phase offset ? – Steve Jun 2 '20 at 11:55
• @Steve yes when you sample at zero IF you will see any offset introduced by all analog sections as a rotation of the constellation. Timing recovery typically cannot track a carrier offset as it usually involves shifting the location of the sampling clock (effectively) in time. You will need a phase rotator to eliminate CFO but this can be done at complex baseband. – Dan Boschen Jun 2 '20 at 11:59