# carrier frequency difference

Papers use wireless device-dependent radio-metrics as fingerprints. A radio-metric is a component of radio signal, amplitude, frequency and bandwidth. Each device creates a unique set of radio-metrics in its emitted signal due to hardware variability during the production of the antennas, power amplifiers, ADC and DAC circuits.As a result, radio-metrics cannot be altered post-production, and thus provide a reliable means for distinguishing wireless devices.

In fact, the transmitted RF signal from a wireless device experiences hardware impairments, channel characteristics, and noise at the receiver. Distortions caused by channel-specific and noise-related effects are likely to have a more random structure. Distortions in a feature that are caused by transmitter hardware impairments should manifest themselves consistently across multiple frames from the same transmitter.

device-dependent fingerprints are Modulation based features like carrier Frequency offset, phase shift offset, I/Q origin offset in the constellation plane.

1. How can I simulate carrier frequency offset in MATLAB?

2. In the following reference 1, the carrier frequency difference (CFD) is used as a feature for distinguishing wireless devices and it is assumed as Gaussian distribution. Why can we assume that this feature is Gaussian distributed?

3. Why do noise and the channel not affect this feature?

On Identifying Primary User Emulation Attacks in Cognitive Radio Systems Using Nonparametric Bayesian Classification

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CFO of different devices in the feature space

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Device fingerprinting to enhance wireless security using nonparametric Bayesian method

1. Communication systems are usually simulated in the equivalent baseband. As a consequence, all signals are generally complex-valued and the carrier frequency is zero. To model a carrier frequency offset multiply the transmit signal with a complex exponential:

f_off = 1e6; % in Hz
x_off = x .* exp(1i * 2*pi * f_off/f_s * 1:numel(x));


Here I've assumed that the real-world analog signals are modeled by oversampling the discrete-time signals. The oversampling rate is f_s.

2. This is probably a pragmatic assumption as Gaussian distributions are understood well and are mathematically relatively convenient. Presumably, it also includes the assumption that higher frequency offsets are less probable (which is quite plausible, I think). In contrast, a uniform distribution would certainly be a bad model. The authors do not motivate their choice of a Gaussian distribution, so they probably haven't done measurements to validate their assumption. But I think as a first approach it's sufficient.

3. Noise is the addition of a random signal. As such it will never shift frequencies but "only" cause a time-variant change of amplitude and phase of the frequencies in the considered bandwidth. Of course, noise will affect the carrier frequency estimation which will get more difficult with increasing noise power, but again it will not shift frequencies.

In contrast, the channel can indeed introduce a frequency shift when the user device moves. This causes the Doppler effect which manifests itself in a frequeny shift of the signal. However, the authors neglect this effect saying that

For OFDM application with low mobility, the Doppler effect is not severe. Moreover, due to reflection, the Doppler effect will cause the carrier to smear on both sides of the frequency. Consequently, the transmitter oscillator is the major factor for this feature.

• In the simulation, for a given wireless device should I consider the carrier frequency offset (CFO) constant or a random variable that is Gaussian distributed? CFO~$Normal(0,sigma^2)$ If the carrier frequency difference for a device is the random variable, It has different values at each time. Then, for distinguishing devices using this feature, we have Gaussian distributions with different variances. Is it right? On the other hand, we can consider a Gaussian distribution with non-zero mean (CFO~$Normal(mean,sigma^2)$) for each device that $mean$=CFO for specified device – Khatereh Feb 13 '15 at 12:26
• To be honest, I'm not sure how the paper should be interpreted here. But as they assume a zero mean, I tend to think that the CFO of the individual device is drawn from a Gaussian distribution. On the other hand, the CFO will change with time due to temperature and aging. – Deve Feb 13 '15 at 18:43
• Thank you for your guidance. I said about Gaussian distribution with non zero mean based on the above image. I don't know maybe my opinion isn't correct. – Khatereh Feb 13 '15 at 19:41
• The distribution should have zero mean as a positive deviation from the ideal frequency is equally probable as a negative deviation. – Deve Feb 13 '15 at 22:25

1 As defined, the device-dependent fingerprint offset should be constant for a given device so if you have nominal_carrier_freq = 1000; then you could say

unique_dev_offset = 1;
unique_dev_carrier_freq = nominal_carrier freq + unique_dev_offset


2 Using the Gaussian distribution is common for modeling any deviation from nominal. Think of the context. Our goal is to have devices that are identical, but in real life they have some slight differences. Take a resistor for example, with a nominal value R. If we have a set of resistors we will see some are a little different than R but the majority of them are centered around R. So our model single modal distribution around R. With decreasing probability of being far away from R. This could be a lot of functions (triangle for instance meets those requirements) but the Gaussian is used because it has nice mathematical properties and usually fits real world data very well

3 I think you have to say the channel is constant for this to work. now imagine we have two signals. both are supposed to be the same but they have some unique characteristics. in the images below, I have two signals, one at nominal frequency (0 carrier freq offset), and one with a nonzero carrier frequency offset. We can see that even after adding noise the signals are still distinguishable from each other. Noise won't affect the characteristic frequency offset of the signals. It may modify the appearance of the signals, but not the underlying characteristics

hope that helps

• Thank you for your help. The noise does not affect the frequency offset of the signals. If suppose modulated signals and their symbols on constellation plane, frequency offset causes phase rotation in the constellation plane. The noise affects on constellation. For low SNR, the effect of noise on constellation is more. Is that correct? – Khatereh Feb 13 '15 at 16:46
• Correct! Just think of what SNR means, the most simplistic definition is SNR = Psignal / Pnoise. As this number gets smaller we are saying the noise is getting larger in comparison to the actual signal. So the affect of noise is now stronger since it is a bigger portion of the signal. This means any noise now plays a bigger role in the output and causes a larger rotation in our constellation – andrew Feb 17 '15 at 22:22