# Design of Shadow Filter Using Low-Voltage Multiple-Input Operational Transconductance Amplifiers
## 1 Introduction  
Universal filters typically provide second-order transfer functions for low-pass filters (LPFs), high-pass filters (HPFs), band-pass filters (BPFs), band-stop filters (BSFs), and all-pass filters (APFs), all within the same topology under suitable conditions. Filters that allow for the electronic and orthogonal control of both the natural frequency and the quality factor are particularly desirable, as they enable the desired natural frequency and quality factor to be achieved without the need to redesign the passive components. The natural frequency and quality factor of universal filters are typically adjustable through internal parameters such as resistors, capacitors, and transconductances [1, 2, 3, 4, 5]. However, adjusting these internal parameters can also alter the performance of the filters, affecting factors like dynamic range, operating frequency, and power consumption. Shadow filters, also known as frequency-agile filters, are techniques that use external amplifiers to adjust the natural frequency and quality factor of conventional universal filters. Two primary techniques for shadow filters have been described in the literature [6; 7]. The first shadow filter concept, presented in [6], involves a second-order filter with low-pass filter (LPF) and band-pass filter (BPF) outputs, along with an external amplifier. The output signals from the LPF and BPF are summed, and this summed output is amplified by an external amplifier before being fed back to the input of the original second-order filter. By adjusting the gain of the amplifier, the quality factor of the LPF and BPF can be modified. However, this method only allows for the adjustment of the quality factor, without providing control over the natural frequency. To address this limitation, the shadow filter concept was further developed in [7] to allow for the modification of both the natural frequency and the quality factor using external amplifiers. In this second approach, the second-order filter includes LPF, high-pass filter (HPF), and BPF outputs, and it utilizes two external amplifiers. The BPF output is amplified by the first external amplifier, while the LPF output is amplified by the second external amplifier. These two amplified output signals are then fed back to the input of the original second-order filter, enabling control over both the natural frequency and the quality factor. Numerous shadow filters have been proposed in the literature, utilizing the concepts from [6; 7] to implement these filters using various active elements [8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34]. For example, the shadow filter in [8] employs second-generation current conveyors, while the shadow filters in [9; 10; 11] make use of current-feedback operational amplifiers. Additionally, the shadow filter in [12] uses operational transresistance amplifiers, and the one in [13] uses operational floating current conveyors. However, in the case of shadow filters using these active devices [8; 9; 10; 11; 12; 13], the natural frequency of the original second-order filters cannot be electronically controlled. The ability to electronically tune the natural frequency of the original second-order filters in shadow filters offers significant advantages, such as compensating for variations in the natural frequency due to temperature and process changes. This electronic tuning capability enhances the robustness and stability of the filter's performance across varying temperature and process conditions. Numerous shadow filters with electronic tuning capabilities have been proposed in the literature [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34]. For example, the shadow filters presented in [14; 15; 16; 17] are designed to deliver high-quality-factor (high-Q) band-pass filters (BPFs).