The mathematical process of switching mixing is thoroughly analyzed, and a detailed implementation method is provided. Both theoretical analysis and practical experiments have demonstrated that the mixing approach based on analog switches can effectively overcome the shortcomings of traditional nonlinear components or multiplier-based mixing techniques. This method eliminates the influence of the local oscillator signal while preserving the parameter information of the input signal to the maximum extent.
*Fund Project: National Natural Science Foundation of China (11602300)
Typically, mixing is achieved using nonlinear components or dedicated multipliers. However, these methods inevitably introduce amplitude and phase information from the local oscillator into the output signal, which is often undesirable. Moreover, both nonlinear components and multipliers tend to generate various types of interference and distortion, such as whistling, parasitic channel interference, cross-modulation distortion, and intermodulation distortion. These issues can severely degrade receiver performance. In contrast, the switching mixing technique can significantly suppress these negative effects, making it a more reliable option for high-quality signal processing.
1. Mixing Principle
A mixing circuit, also known as a frequency converter, plays a crucial role in superheterodyne receivers by enabling undistorted spectrum shifting [1]. There are multiple ways to achieve mixing, with the most common being the use of a multiplying circuit. This can be implemented using either a nonlinear device or a dedicated integrated circuit multiplier.
Assuming the two input signals to the multiplier are:
Vâ‚ = A cos(ωâ‚t + φâ‚),
V₂ = A cos(ω₂t + φ₂),
Then the output of the multiplier is:
Vo = AB/2 [cos((ω₠+ ω₂)t + (φ₠+ φ₂)) + cos((ω₠- ω₂)t + (φ₠- φ₂))] (1)
In the context of a receiver, only the low-frequency component of the signal is usually of interest. Therefore, after applying a first-stage low-pass filter, the output can be expressed as:
Vo = AB/2 cos((ω₠- ω₂)t + (φ₠- φ₂)) (2)
It is clear that the amplitude, phase, and frequency of the output signal depend on both input signals. For a receiver, the focus is on how the received echo signal affects the output, rather than introducing too many parameters from the local oscillator. Additionally, since multipliers are nonlinear devices, they can produce significant interference and distortion, especially when the input signal amplitude increases. This can lead to signal saturation and distortion [3], which negatively impacts receiver performance. Hence, using a multiplier is not always the best choice for implementing a mixing circuit. Switching mixing offers an effective alternative that overcomes these limitations.
2. Switching Mixing Principle
Assume the input signal and the local oscillator signal are:
Vâ‚ = A cos(ωâ‚t + φâ‚),
V₂ = A cos(ω₂t + φ₂).
Now consider a square wave signal v'â‚‚ with the same frequency as vâ‚‚. Expanding this into a Fourier series yields:
v'₂ = 4/π [cos(ω₂t) + 1/3 cos(3ω₂t) + 1/5 cos(5ω₂t) + ...]
This shows that v'₂ consists of multiple harmonics, with ω₂ as the fundamental frequency [4]. Therefore, instead of directly mixing v₠and v₂, we can mix v₠with v'₂ and then extract the fundamental frequency component to obtain the result of mixing v₠and v₂.
The mixing of vâ‚ and v'â‚‚ can be written as:
v_out = v₠× v'â‚‚ = A cos(ωâ‚t + φâ‚) × [4/Ï€ cos(ω₂t) + 4/3Ï€ cos(3ω₂t) + ...]
After filtering out higher harmonics using a low-pass filter, the resulting output is:
v_out = (4A/π) cos((ω₠- ω₂)t + (φ₠- φ₂)) (6)
From equation (6), it is evident that the final output depends only on the input signal vâ‚, and the frequency corresponds to the difference between the input signal and the local oscillator signal. This means that the switching mixing method preserves the original signal information while minimizing the impact of the local oscillator signal.
3. Implementation of the Switching Mixer
Based on the above principle, the mixing of vâ‚ and v'â‚‚ can be expressed as follows:
v_out = v₠× v'₂
It can be observed that during the positive level of v'â‚‚, the output is vâ‚, and during the negative level, the output is -vâ‚. This behavior can be achieved using an analog switch, where the control signal is the square wave v'â‚‚ [5].
The specific implementation involves splitting the input signal vâ‚ into two paths: one is vâ‚, and the other is -vâ‚. Two analog switches are used to control the output. When v'â‚‚ is at a positive level, the upper switch is turned on, and vâ‚ is output. When v'â‚‚ is at a negative level, the lower switch is activated, and -vâ‚ is output. This setup matches the expected result from equation (7), effectively achieving the multiplication of vâ‚ and the square wave v'â‚‚.
The waveform of the output signal is shown in Figure 2. It is important to note that the output contains higher harmonics, so a low-pass filter is required to extract the desired difference signal.
4. Conclusion
As demonstrated through the analysis, the switching mixing technique can replicate the function of a multiplier while ensuring that the output signal's amplitude and phase are solely dependent on the input signal. This makes it more suitable for subsequent signal detection and parameter identification processes.
For multi-input receiver systems, the switching mixing method is particularly beneficial in reducing parameter distortions caused by inconsistencies in the local oscillator signal. This improves the receiver's ability to recognize and process the target signal effectively, making it a promising approach for modern communication systems.
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