Chopping: a technique for noise and offset reduction
5.4. Noise modulation
From the previous paragraph we have seen the modulation effects of chopping on a input process x(t). It is important to notice the difference between a narrow-band process and a broadband process after chopper modulation. The input stationary random process x(t) can be white noise or 1/f noise applied to the amplifier A(f) which has a band limiting effect on the noise.
5.4.1. White noise modulation
For the beginning, assume that x(t) is a broadband random process white noise like with a power spectral density S_{white}. In order to simplify the analysis, the gain of the amplifier is taken to be 0dB and the frequency transfer has a first order behavior:
(5.12)
The power spectral density of the noise at the output can be found from (5.11) with assumption that S_{xx}(f)=S_{white}:
(5.13)
The series from (5.13) can be computed by using Poisson summation rule [9]. For large values of f_{LP}T (f_{LP}T>>1) we get:
(5.14)
In conclusion, the chopper modulator has a small influence on the white noise when the bandwidth of the amplifier is larger than the chopping frequency. This is not the case for sampling where undersampling phenomena actually increases the noise in the baseband [1]. The power spectral density of the white noise will be unchanged as long as the bandwidth of the amplifier is larger than the chopping frequency. Chopping at frequencies higher than f_{LP} will reduce the power spectral density of the white noise as explained in reference [4]. Although white noise is a fundamental limitation it can be reduced by chopping. Oversampling in D/A and A/D converters has about the same effect on the baseband white noise.
5.4.2. 1/f noise modulation
When narrow band random processes are applied at the input of the chopper modulator the situation will change. The input power spectral density of 1/f noise is:
(5.15)
In the constant c we have included the 1/f noise constant k_{1/f} process dependent and the geometry factor, dependent on the dimensions of the transistors. After chopping the PSD of the modulated 1/f noise becomes:
(5.16)
Fig.5.5 shows the normalized PSD of the flicker noise after chopping when f_{LP}T>>1. The effect of chopper modulation on the flicker noise will be the reduction of the PSD of the output noise at low frequencies. At odd multiples of the chopper frequency the
Fig.5.5: PSD of 1/f noise after chopper modulation |
PSD of the flicker noise increases. Thus, under the same condition f_{LP}T>>1 the PSD of the white noise after chopping remains the same and the PSD of the flicker noise at low frequencies will be reduced.