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Complex Frequency Shifting in Continuous Time
December 1, 2022

Table of Contents

Introduction

Complex frequency shifting moves the frequency response of a signal in the frequency domain. This blog describes complex frequency shifting in continuous time by multiplying a signal with a complex exponential.

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Complex Frequency Shifting in the Frequency Domain

Complex frequency shifting is the process of changing the location of a signal in the frequency domain. For a signal x(t) the frequency response is defined as X(f), related through the Fourier transform

(1)   \begin{equation*}X(f) = \mathcal{F} \{ x(t) \}.\end{equation*}

Complex frequency shifting by frequency f_c moves the frequency response within the frequency domain. Mathematically, complex frequency shifting in the frequency domain is represented by substituting

(2)   \begin{equation*}f \rightarrow f - f_c\end{equation*}

such that the frequency response is now

(3)   \begin{equation*}X\left(f - f_c\right).\end{equation*}

Figure 1 gives an example of how X(f) can be frequency shifted by frequency f_c.

Figure 1: Complex frequency shifting in the frequency domain.
Figure 1: Complex frequency shifting in the frequency domain.

Complex Frequency Shifting in the Time Domain

Complex frequency shifting in the time domain is accomplished by multiplying a signal x(t) with a complex sinusoid,

(4)   \begin{equation*}y(t) = e^{j2\pi f_c t}\end{equation*}

resulting in

(5)   \begin{equation*}z(t) = x(t) \cdot y(t).\end{equation*}

The Fourier transform of (5) is defined by (reference)

(6)   \begin{equation*}\begin{split}Z(f) & = \mathcal{F} \{ x(t) \cdot y(t) \} \\& = X(f) \ast Y(f)\end{split}\end{equation*}

where \ast represents linear convolution. The Fourier transform of (4) is defined as (reference)

(7)   \begin{equation*}Y(f) = \delta \left( f - f_c \right).\end{equation*}

Substituting (7) into (6),

(8)   \begin{equation*}Z(f) = X(f) \ast \delta \left(f - f_c\right)\end{equation*}

which can be simplified asĀ 

(9)   \begin{equation*}Z(f) = X\left(f - f_c\right)\end{equation*}

from the Dirac delta sifting theorem (reference).

Conclusion

God, the Lord, is my strength; he makes my feet like the deer’s; he makes me tread on my high places. Habakkuk 3:19

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