3.2.4 Effect of integrator non-idealities in filter
3.2.4.4 Dynamic range performance
denormalized to the corresponding value. Thus, the integrated output noise power would still be the same over the frequency tuning range. This effect can also be verified by using the linear time scaling technique [37].
3.2.4.4 Dynamic range performance
For analog circuits, some of different performances should meet the system specification. The typical filter performances are the accuracy of the transfer function, power consumption, linearity and noise. The linearity would limit the largest input signal swing range and the noise limit the smallest useful values. Thus, the dynamic range would be determinate by the linearity and the noise together.
3.2.4.4.1 Total harmonic distortion (THD)
For linear time-invariant system, the output signal would be linearity related to the input signal. The output signal would have the same frequency component but the magnitude and phase could be different. If the input signal at specific frequency is applied to non-linear time-invariant system, the output signal would include additional harmonic components. The total harmonic distortion (THD) is defined to be the ratio of the power including second and higher order harmonic terms to the power of the first order component of the signal. It has the following relation:
2 2
harmonic component. The THD can also be presented as a percentage value. Since the THD is given by a ratio, the value should be reported with the combination of an input signal level. In practical, the first five to ten harmonics are calculated owing to that high order terms would be less than the noise power. A THD with the value of -40dB, which is equivalent to one percent (1%), would be usually required for a Gm-C implementation.
For a non-linear circuit, we apply a sinusoid signal Acos(ωt) to input terminal and then the output can be expanded by a Taylor series formula and it exhibits
2 3
Since the third-order harmonic term dominates the linearity performance of the differential circuit, we neglect even order distortion component. (3.41) can be third-order distortion term. Usually, (3/4)V3A3 << VfA can be hold and HD3 is given by should be less than 1/3 of cutoff frequency. If an input signal with higher frequency is applied, the harmonic components would appear at filter stopband, and thus the obtained value is attenuated by the filter. However, the parasitic capacitance would degrade the linearity of transconductor at higher frequency. Then, the filter linearity is also reduced and the filter is worse at highest frequency. The highest frequency, which interests in signal processing of the filter, is the cutoff frequency. To obtain the
actual linearity performance of the filter, we should measure the linearity performance at cutoff frequency and the two-tone test described in next section is more suitable.
The THD is a straight and simple test but it would not work well with signals near passband.
3.2.4.4.2 The third-order intercept point (IP3)
The third-order intercept point(IP3) is a measure for third-order distortion component.
Before we present the IP3, the two-tone inter-modulation test should be introduced at first. By using two-tone signals with different frequencies to the input of a circuit, the output signal results from the multiplied input signals. The property can be obtained by given a two-tone signal:
1 1 2 2
( ) cos( ) cos( )
v tin =A ωt +A ω t (3.44) As the non-linear output expanded by a Taylor series formula in (3.41), the output signal in this case can be given by
[ ] [ ]
also called the third-order inter-modulation (IM3). The first and second line of (3.45) shows the fundamental component. The third and fourth line of (3.45) shows the distortion at nearly three times the fundamental frequency and two new frequencies that are close to the input frequencies. Thus, the distortion terms can appear at frequencies near the input signal if the frequencies of two-tone signal are very close to each other. The IM3 can be given asFigure 3.12. The illustration of input and output intercept point with the unit of decibel.
2
3 3
3
3
IM 4
f f
A V A
A = A = V (3.46)
where Af is the amplitude of fundamental component and A3 is the amplitude of inter-modulation term. The measurement is suitable for narrow band applications such as a band-pass filter because the distortion terms would appear at passband. This test can also be used to obtain the performance for a low-pass filter at cutoff frequency.
Since IM3 is result of a ratio, we should give the input signal power at the same time to characterize the circuit behavior. On the other hand, the IP3 shows another way. It can be measured when the magnitude of the input signal is small, and thus the higher order distortion terms are negligible. If we neglect the cubic distortion term of the first and second line in (3.45), we can obtain that the magnitude of fundamental term would be proportional to the magnitude of input signal. For the third and fourth
line, the magnitude of inter-modulation terms has a cubic function. Thus, the magnitude of inter-modulation term would grow at three times than fundamental terms in a logarithmic scale, and the IP3 is defined to be the intersection point of the two lines. Fig. 3.12 shows the logarithmic illustration of the IP3. The horizontal coordinate of the point is called the input IP3 (IIP3), and it is the input signal magnitude when the third-order harmonic term equals to the fundamental terms. The vertical coordinate is called the output IP3 (OIP3). In a low-pass filter design with unity gain transfer function, the IIP3 would equal to OIP3.
The IP3 not only shows the power of input signal but also shows the circuit
Then, the OIP3 is equal to VfAIP3. To obtain the IP3 in practical, small input signal is applied to point out the line of third-order distortion component. The reason in that the IP3 is usually out of allowable input range and the value is sometimes larger than the supply voltage in nano-scale CMOS technology. Even though a large input signal can be applied, the higher order non-linearity term would appear and the compression effect occurs.
A useful method to measure IIP3 is discussed. By substituting (3.46) into (3.47), we can obtain 1/AIM3 = AIIP32
/A2. In the logarithmic scale, the value can be expressed as to half the absolute value of IM3 plus the corresponding input level. Thus, the IP3 can be measured with only one input level rather than extrapolation.
3.2.4.4.3 Spurious-free dynamic range (SFDR)
The spurious-free dynamic range is defined to be the signal-to-noise ratio when the third-order inter-modulation components equal to the noise power. This condition
happens at the small input signal amplitude. The relationship can be expressed to SFDR = Af*-No = Af*- A3*, where No is the noise power, Af* and A3* are the magnitudes of output and inter-modulation component when inter-modulation component is equal to noise power. Using (3.48), we can obtain