Linear Frequency Invariant (LFI) Systems

In this post, as opposed to our usual notation of a capital letter for a frequency domain signal, e.g., S(F), we introduce a small letter like s(F) instead to highlight the fact that the system is linear frequency-invariant, and not linear time-invariant.

The linear part in a Linear Frequency-Invariant (LFI) system is defined as before. For an input s(F) generating an output r(F), a linear system implies that if two inputs are scaled and summed together to form a new input, the new output of the system is also a scaled sum of their individual outputs.

$$
\text{input}\quad \alpha_1 s_1(F) + \alpha_2 s_2(F) ~ \xrightarrow~ \alpha_1 r_1(F) + \alpha_2 r_2(F) \quad \text{output}
$$

On the other hand, a frequency-invariant system implies that if an input signal is shifted in frequency by some amount F_0, the output also just gets shifted in frequency by F_0 and remains exactly the same otherwise.

$$
\text{input}\quad s(F-F_0) ~ \xrightarrow~ r(F-F_0) \quad \text{output}
$$

Just like an LTI system, convolution in frequency domain is equivalent to multiplication in time domain in an LFI system as well. However, the main distinction is the following.

The output of an LTI system is the convolution of its impulse response with the input in time domain and multiplication of its frequency response with that of the input in frequency domain. On the other hand, the output of an LFI system is the convolution of its frequency response with that of the input in frequency domain and multiplication of its impulse response with the input in time domain.

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