Frequency-Domain Response of Lock-in Filters

Introduction 

The response of a lock-in amplifier is characterized by the parameters of its low-pass filter (LPF), i.e. time constant and filter order. The temporal response of the LPF demonstrates the latency of measurement while its spectral response shows the noise characteristics of measured signals. In the blog post Time-Domain Response of Lock-in Filters [1], we can see how to measure the temporal response of the demodulation filters using the DAQ module (formerly SW Trigger) of LabOne user interface. In the current blog, we will learn how to obtain the frequency response [2] of these filters using the Sweeper module of LabOne. We also present analytical formulas for two types of filter bandwidth in terms of filter order and time constant. 

Theoretical Background 

An \(n\)-th order demodulator filter of lock-in amplifiers consists of \(n\) cascaded first-order low-pass filters. Therefore, the frequency response of the filter is obtained by \(n\)-fold multiplication of the first-order filter response. For the time constant \(\tau\), the frequency response of the \(n\)-th order filter is given by the following expression:

\[H(\omega)=\frac{1}{(1+i\omega\tau)^n} \]

The following figure shows in a linear scale the frequency response of low-pass filters with \(\tau=69.23\ \mu s\) and an order up to \(8\). 

Fig. 1. Frequency response of a low-pass filter with a fixed time constant at several filter orders from 1 to 8 in a linear scale.


For such a filter, it is possible to define two kinds of bandwidth, i.e. 3-dB cut-off bandwidth and noise equivalent power (NEP) bandwidth. They can be expressed in terms of time constant \(\tau\) and filter order \(n\). The 3-dB cut-off bandwidth \(\omega_\text{3dB}\) is defined as the frequency at which the filter amplitude is 3 dB less than its maximum following the equation below: 

\[|H(\omega_\text{3dB})|^2=\frac{1}{2}\]

Using the above expression, the 3-dB cut-off bandwidth of the \(n\)-th order filter in Hz is obtained as follows considering the relation \(\omega_\text{3dB}=2\pi f_\text{3dB}\). 

\[f_\text{3dB}^{(n)}=\frac{\sqrt{2^{\frac{1}{n}}-1}}{2\pi\tau},\ \ \ \ \ \ \ \ n=1,\ 2,\ 3,…\]

On the other hand, the NEP bandwidth \(\omega_\text{NEP}\) is defined as a bandwidth of a corresponding rectangular filter with identical power. Therefore, the NEP bandwidth of the normalized \(H(\omega)\) is obtained as follows: 

\[\omega_\text{NEP} = \int_{0}^{\infty}{|H(\omega)|^2 d\omega}\]

The NEP bandwidth can be expressed in Hz instead of rad/s using the simple relations \(\omega_\text{NEP}=2\pi f_\text{NEP}\). The above expression results in the following recursive relation for the NEP bandwidth of the \(n\)-th order filter. 

\[\begin{align} &f_\text{NEP}^{(n+1)} = \frac{2n-1}{2n}f_\text{NEP}^{(n)}, \ \ \ \ \ \ \ \ n=1,\ 2,\ 3,… \\ &f_\text{NEP}^{(1)} =\frac{1}{4\tau} \end{align}\]

The above recursive relation can be solved to obtain an explicit formula for the NEP bandwidth of the \(n\)-th order filter as follows. 

\[ f_\text{NEP}^{(n)} = \frac{(2(n-1))!}{(2^n (n-1)!)^2} \frac{1}{\tau}, \ \ \ \ \ \ \ \ n=1,\ 2,\ 3,…\]

The following table shows the 3-dB and NEP bandwidths of low-pass filters up to order 8 expressed in terms of time constant inverse \(1/\tau\) which are used for demodulation in lock-in amplifiers made by Zurich Instruments. 

\(\ n\) \(\ 1\) \(\ 2\) \(\ 3\) \(\ 4\) \(\ 5\) \(\ 6\) \(\ 7\) \(\ 8\)
\(f_\text{3dB}^{(n)}\ \ \ \ (1/\tau)\) \(\ 0.1592\) \(\ 0.1024\) \(\ 0.0811\) \(\ 0.0692\) \(\ 0.0614\) \(\ 0.0557\) \(\ 0.0513\) \(\ 0.0479\)
\(f_\text{NEP}^{(n)}\ \ \ (1/\tau)\) \(\ 0.2500\) \(\ 0.1250\) \(\ 0.0938\) \(\ 0.0781\) \(\ 0.0684\) \(\ 0.0615\) \(\ 0.0564\) \(\ 0.0524\)

 

Using the above table one can easily calculate the 3-dB and NEP bandwidths of an \(n\)-order LPF in Hz based on the filter time constant \(\tau\) in s. It is also evident that the NEP bandwidth is larger than the 3-dB bandwidth and as the filter order increases, the two bandwidths merge together.  

Experimental Measurement 

In this section, we learn how to measure the frequency response of low-pass filters in lock-in amplifiers using the sweeper module of LabOne user interface. The idea behind this technique is to apply a fixed frequency to the signal input and to sweep the frequency of demodulator in the vicinity of the drive frequency. This measurement can be done by any of lock-in amplifier made by Zurich Instruments as the sweeper module is available for all instruments. In this measurement, a 5-MHz MFLI Lock-in Amplifier equipped with the MF-MD Multi-demodulation option is used to obtain the frequency response of the lock-in filters. Figure 2 shows the proper settings of the instrument and the sweeper module. A signal at 1 MHz is generated from oscillator 2 at the signal output and looped back to the signal input. The frequency of oscillator 1 is swept in the vicinity of 1 MHz by the sweeper tool and the amplitude of input signal is measured. The filter time constant is fixed at \(\tau=69.23\ \mu s\) while each measurement is carried out for a filter order from 1 to 8.   

Fig. 2. Proper settings of instrument and sweeper as well as 8 measured filter responses for filter orders from 1 to 8; (click to enlarge).


Fig. 2 shows the measured frequency response of 8 filters with an identical time constant but 8 different filter orders. The measurements in Fig. 2 look similar to the theoretical results in Fig. 1. To verify it, we can plot both measured responses and analytical curves together in a logarithmic scale. Fig. 3 depicts the filter response obtained by measurement and calculated from the analytical formula which indicates a perfect agreement between theory and experiment. 

Fig. 3. Frequency response of demodulation filter for 8 orders obtained by theory and measurement. As the order increases from 1 to 8, the filter slope decreases from -6 to -48 dB/octave.

It is evident from Fig. 3 that for a given time constant \(\tau\), the 3-dB bandwidth and thus the NEP bandwidth will reduce by increasing the filter order. Moreover, the filter response becomes sharper as the filter order increases. These characteristics should be kept in mind when a proper measurement configuration is being set-up. For further details regarding the lock-in measurement, one can refer to the Zurich Instruments white paper on the principles of lock-in detection [3].  

Conclusion

In this blog, two analytical formulas for 3-dB and NEP bandwidths are given in terms of filter order and time constant. Moreover, a technique to measure the filter response is proposed. This method uses the sweeper module of LabOne to obtain the frequency response of the low-pass filters deployed in lock-in amplifiers.    

References

  1. M. Alem, “Time-Domain Response of Lock-in Filters.” 
  2. Wikipedia: “Frequency response.”
  3. Zurich Instruments: “Principles of lock-in detection and the state of the art.”