Wave Walker DSP

DSP Algorithms for RF Systems

Trending

Buy the Book!

DSP for Beginners: Simple Explanations for Complex Numbers! The second edition includes a new chapter on complex sinusoids.

bookCover2Ed
examples1
examples2
eulersFormula
radians
negativeFrequency
adding
posNegAngles
complexSinusoidPhase
Complex Frequency Shifting in Discrete Time
January 1, 2023

Table of Contents

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.

Check out these other blogs!

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.

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

Figure 2: Complex frequency shifting can result in aliasing if the frequency response overlaps the half sampling rate.
Figure 2: Complex frequency shifting can result in aliasing if the frequency response overlaps the half sampling rate.

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).

Check out these other blogs!

Leave a Reply

For everything there is a season, and a time for every matter under heaven. A time to cast away stones, and a time to gather stones together. A time to embrace, and a time to refrain from embracing. Ecclesiastes 3:1,5
The earth was without form and void, and darkness was over the face of the deep. And the Spirit of God was hovering over the face of the waters. Genesis 1:2
Behold, I am toward God as you are; I too was pinched off from a piece of clay. Job 33:6
Enter His gates with thanksgiving, and His courts with praise! Give thanks to Him; bless His name! Psalm 100:4
Lift up your hands to the holy place and bless the Lord! Psalm 134:2
Blessed is the man who trusts in the Lord, whose trust is the Lord. He is like a tree planted by water, that sends out its roots by the stream, and does not fear when heat comes, for its leaves remain green, and is not anxious in the year of drought, for it does not cease to bear fruit. Jeremiah 17:7-8
He said to him, “You shall love the Lord your God with all your heart and with all your soul and with all your mind. This is the great and first commandment. And a second is like it: You shall love your neighbor as yourself. On these two commandments depend all the Law and the Prophets.” Matthew 22:37-39
Then He said to me, “Prophesy over these bones, and say to them, O dry bones, hear the word of the Lord. Thus says the Lord God to these bones: Behold, I will cause breath to enter you, and you shall live." Ezekiel 37:4-5
Riches do not profit in the day of wrath, but righteousness delivers from death. Proverbs 11:4
The angel of the Lord appeared to him in a flame of fire out of the midst of a bush. He looked, and behold, the bush was burning, yet it was not consumed. And Moses said, “I will turn aside to see this great sight, why the bush is not burned.” When the Lord saw that he turned aside to see, God called to him out of the bush, “Moses, Moses!” And he said, “Here I am.” Exodus 3:2-3
Daniel answered and said: “Blessed be the name of God forever and ever, to whom belong wisdom and might. He changes times and seasons; He removes kings and sets up kings; He gives wisdom to the wise and knowledge to those who have understanding." Daniel 2:20-21
Now the Lord is the Spirit, and where the Spirit of the Lord is, there is freedom. 2 Corinthians 3:17
Previous slide
Next slide

This website participates in the Amazon Associates program. As an Amazon Associate I earn from qualifying purchases.

© 2021-2024 Wave Walker DSP