#### 3.1. Clock Noise Model

[10] The output of an ideal clock can be expressed as a sine wave with constant amplitude, carrier frequency, and phase reference. Its spectrum is an impulse at the carrier frequency. For a practical (nonideal) clock, the output is more generally given by [*Hajimiri and Lee*, 1999]

where *V*_{0} is the maximum voltage swing, *n*(*t*) is amplitude fluctuation, *f*_{0} is the center frequency, and θ(*t*) is the phase fluctuation of the signal. *V*_{0} and *f*_{0} represent the shape of the ideal clock output waveform. The spectrum of a practical clock has sidebands close to the center frequency, *f*_{0}, and to its harmonics. The sidebands are generally referred to as *phase noise* in the frequency domain.

[11] Because of amplitude-limiting effects, the amplitude noise in an oscillator output is significantly reduced and can generally be ignored [*Hajimiri and Lee*, 1999]. Furthermore, because phase fluctuations can produce a random frequency, which is the derivative of the random phase fluctuation with respect to time, the phase fluctuation θ(*t*) can instead be represented by a frequency modulated component, or

where *f*_{Δ} represents the maximum frequency shift away from *f*_{0} in one direction and *x*(*t*) is the frequency modulating signal. Equation (2) assumes that *x*(*t*) is limited to the range ±1. The signal *x*(*t*) is usually a random signal and represents frequency modulation noise in the oscillator. However, it might also contain a deterministic component. For example, an unwanted nonrandom signal might leak into an oscillator circuit or a signal might be added intentionally. In this article, we intentionally introduce sinusoidal frequency modulation signals, *x*(*t*), into a modified clock circuit to investigate their effect on digital radiometer performance.

[12] Using the clock model described by equation (2), the clock frequency stability can be defined as

where 〈·〉 denotes expectation in the case of a random signal and the average over one period in the case of a sinusoidal signal.

#### 3.2. Linear Tap Delay Line Filter Response to Clock Noise

[13] The digital back end includes a bank of FIR filters that divide the complete signal bandwidth into narrow subbands. Each of the subband filters can be represented as [*Oppenheim et al.*, 1983]

and its Fourier transform is

where {*b*_{n} ∣ *n* = 0, 1, …, *N*} is the set of the filter coefficients and Ω is the normalized frequency. Each individual subband filter has its own set of coefficients.

[14] Instability in the clock frequency can cause variations in the sample frequency, the bandwidth, and the center frequency of the subband filters. It results in a band-pass filter “shaking” in the frequency domain. A signal in the rejection region of a filter will have more chance to enter the passband of the filter. The degree of leakage will depend on the statistical distribution of the clock frequency.

[15] To simplify the analysis, we normalize the frequency of the input signal to that of the sample frequency. For example, for a sinusoidal input signal with frequency *f*, its normalized frequency after down-sampling will be

where *N* is the down-sampling factor, *f*_{ck,0} is the nominal clock frequency, and *f*_{d} is the frequency deviation of the clock. For the digital back end module considered here, *N* = 20. The instantaneous clock frequency is referred to as *f*_{ck,m}, and the instantaneous frequency deviation is *f*_{d,m} (= *f*_{ck,m} − *f*_{d,m}). The frequency response of the subband filter stage, including the effects of instantaneous changes in clock frequency, will be given by

The output power of the subband FIR filter is determined as the expected value of the filter response, weighted with respect to the probability distribution of the clock frequency deviation. The final expression is given by

The solution to equation (7) can be obtained numerically. Using this approach, the out-of-band rejection of the subband filter can be characterized if the statistical distribution of the clock frequency is known.

#### 3.3. Complex Correlation Response to Clock Noise

[16] An ADC is a device that converts continuous signals to discrete digitals at a rate set by its clock, and it can be treated as a mixer in analog circuits except that the output of the ADC is discrete. The spectrum of the continuous signal differs from that of discrete samples in that the discrete sample spectrum is periodic with a frequency *f*_{s}, the sample frequency. If the ADC clock is not stable, the spectrum of the ADC output is broadened, and aliasing is induced. The width of the spectrum and aliasing increases with increased instability of the ADC clock.

[17] The band-pass filters in each subband are flat in magnitude and linear in phase. Therefore, they will not change the correlation coefficient of the signals if the signals are within the passband. The output of the band-pass filter is split into two paths. One path is simply delayed and is used as the real part of the signal, whereas the other is passed through a Hilbert transform filter to form the imaginary part of the signal. The impulse response corresponding to an ideal Hilbert transform filter in the frequency domain is odd-symmetric, and it has interleaved zeros. Because there is a step function in the frequency domain at DC, the impulse response in the time domain is infinite. When realized in practice, the impulse response must be truncated to a finite length. For example, the digital radiometer considered here uses is a 27-tap approximation. Its impulse response is shown in Figure 2a and its frequency response is shown in Figure 2b.

[18] As Figure 2b shows, the phase response of the truncated Hilbert transform filter is ideal, but its magnitude response is rippled in the frequency domain. It is this ripple that can cause magnitude imbalances between the in-phase and quadrature components of the signal when the clock is not stable. Variations in the magnitude imbalance, due to clock noise, can lead to the changes in the complex correlation.

[19] The impact of clock noise on the complex correlation can be assessed by considering the case of a sinusoidal radiometer input signal. Assume the input signal to the radiometer digital back end is sinusoidal with frequency, *f*, and phase angle, ϕ, between the v- and h-pol inputs. The I and Q filter outputs represent the real and imaginary components, respectively, of the complex signals to be correlated. They can be expressed as

where *ξ* and *η* are the complex signals in the v- and h-pol channels, *F* is a scale factor representing the amplitude balance (or gain balance) between I and Q filters (*F* = 1 with an ideal Hilbert transform filter), and it is the ratio of the I channel gain to the Q channel gain. In the plots of theoretical prediction shown later, the value of *F* is calculated assuming a stable clock. The function *H*(*f*) is the frequency response of the Hilbert transform filter. After leaving the I and Q filters, the complex signals are cross-correlated (i.e., multiplied and averaged). The normalized complex correlation coefficient between the v- and h-pol signals is given by

where 〈·〉 denotes a time average and Ω, the normalized frequency after down-sampling, is given by equation (5).

[20] If the magnitude of the frequency response of the Hilbert transform filter is ideal, or flat, the product of 〈∣*H*(*f*)∣〉 and the scale factor *F* will be 1 and equation (9) reduces to

In this case, the phase of the complex correlation coefficient is ϕ and the magnitude is unity, independent of ϕ. On the other hand, if the Hilbert transform filter is not ideal and the magnitude of its frequency response is not constant versus frequency, then the factor 〈∣*H*(Ω)∣〉*F* will not equal 1 and will vary with frequency. In this case, we have (see Appendix for proof)

If the inequality in equation (11) holds, then a more general expression for the magnitude and phase of the complex correlation coefficient is given by

Comparing equations (10) and (12), the change of the complex correlation coefficient is given by

where *C* is a factor given by

The sensitivity to clock noise of the errors in magnitude and phase of the complex correlation coefficient will vary, depending on the phase angle itself. This can be seen by examining the derivative of equation (13) with respect to phase angle. Maximum sensitivity to noise occurs when the phase angle satisfies

where ϕ_{p,mag} is the phase angle at which the magnitude of the complex correlation coefficient is most sensitive to clock noise and ϕ_{p,pha} is the phase angle at which the phase of the complex correlation coefficient is most sensitive. Over the range [−180, 180], the solution for ϕ_{p,mag} is ±90°. For ϕ_{p,pha}, there are four solutions, given by

Assuming the clock frequency instability is not serious, *C* will be close to 1, and the solutions for ϕ_{p,pha} are ±45° and ±135°.

[21] To illustrate the effect of clock frequency instability on the correlation measurements, two numerical examples are considered. The first example assumes that the clock frequency is varied sinusoidally; the second example assumes that the clock frequency is varied in a random Gaussian manner. The correlation coefficient magnitude and phase variations with clock frequency instability are shown in Figure 3. From these plots, it can be seen that the sensitivity of the correlation coefficient is indeed related to the phase angle of the correlation. If the input phase angle is equal to 0° or 180°, the sensitivity is 0; otherwise, the correlation coefficient magnitude has the highest sensitivity at input phase angle 90° (also at −90°; not shown in Figure 3), whereas the correlation coefficient phase has the highest sensitivity near 45° and 135° (also at −45° and −135°; not shown here).

[22] Note that, with sinusoidal perturbations in clock frequency, the sensitivity of the correlation coefficient has two local maxima as the RMS clock instability is increased, whereas with random Gaussian perturbations the sensitivity grows monotonically with RMS clock noise. This is due to the distribution of the clock frequency in the sinusoidal case. In both cases, however, the phase angles at which sensitivity to noise is maximum are the same and are as predicted by equations (15) and (16).