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Complex Frequency Shifting in Continuous Time

December 1, 2022

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 $f_c$ .

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 as

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

from the Dirac delta sifting theorem (reference).

Conclusion

Multiplication of $x(t)$ with $e^{j2\pi f_c t}$ shifts the frequency response of $X(f)$ by $f_c$ resulting in $X\left(f-f_c\right)$ .

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