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Complex Frequency Shifting in Discrete Time

January 1, 2023

Introduction

Complex frequency shifting is the process of shifting a signal’s frequency response in the frequency domain. This blog describes how to apply complex frequency shifting in the time domain by multiplication with a complex sinusoid.

A previous blog post described complex frequency conversion in continuous time.

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Complex Frequency Conversion in the Frequency Domain

Complex frequency shifting, or frequency conversion, shifts the frequency response of a signal in the frequency domain. A discrete-time signal $x[n]$ has a frequency response $X\left( e^{j\omega} \right)$ related through the Fourier transform,

(1) $\begin{equation*}X\left(e^{j\omega}\right) = \mathcal{F} \{ x[n] \}.\end{equation*}$

Frequency shifting a frequency response by $\omega_c$ is accomplished by substitutting

(2) $\begin{equation*}\omega \rightarrow \omega - \omega_c.\end{equation*}$

such that the frequency response is now

(3) $\begin{equation*}X\left(e^{j\left(\omega-\omega_c\right)}\right).\end{equation*}$

Figure 1 demonstrates how complex frequency shifting by $\omega_c$ appears in the frequency domain.

Discrete-time is only able to represent frequencies

(4) $\begin{equation*}-\pi \le \omega < \pi,\end{equation*}$

where $\pi$ is equal to the half sampling rate in radians. Frequency shifting past the half sampling rate can result in aliasing as shown in Figure 2.

Complex Frequency Conversion in the Time Domain

One method of performing frequency shifting is by multiplying the discrete time signal $x[n]$ by a complex sinusoid

(5) $\begin{equation*}y[n] = e^{j \omega_c n}\end{equation*}$

resulting in

(6) $\begin{equation*}z[n] = x[n] \cdot y[n].\end{equation*}$

The Fourier transform of (6) is defined by [Oppenheim1999, p.59]

(7) $\begin{equation*}\begin{split}Z\left( e^{j\omega}\right) & = \mathcal{F} \{ x[n] \cdot y[n] \} \\& = X\left( e^{j\omega}\right) \circledast Y\left( e^{j\omega}\right)\end{split}\end{equation*}$

where $\circledast$ represents circular convolution. The Fourier transform of (5) is defined as [Oppenheim1999, p.62]

(8) $\begin{equation*}Y\left( e^{j\omega}\right) = 2 \pi \delta \left( \omega - \omega_c \right).\end{equation*}$

Substituting (8) into (7),

(9) $\begin{equation*}Z\left( e^{j\omega}\right) = X\left( e^{j\omega}\right) \circledast 2 \pi \delta \left( \omega - \omega_c \right)\end{equation*}$

which can be simplified as

(10) $\begin{equation*}Z\left( e^{j\omega} \right) = 2\pi X\left( e^{j\left( \omega - \omega_c \right)}\right)\end{equation*}$

from the Dirac delta sifting theorem [Oppenheim1999, p.143].

Conclusion

Multiplication of $x[n]$ with $e^{j\omega_c n}$ shifts the frequency response of $X\left(e^{j\omega}\right)$ by $\omega_c$ , resulting in $X\left(e^{j\left(\omega - \omega_c\right)}\right)$ .