# Wave Walker DSP

## Trending

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

Fourier Transform Pairs of Conjugation and Time Reversal
November 17, 2021

#### Introduction

Too often I find myself asking the questions “what is the Fourier transform of ”? “What is the inverse Fourier transform of ?” Finding them in a book or on Wikipedia often takes too long so to save myself the time and hopefully be useful to others, I’ve provided the derivations for the Fourier transforms of x(t), , x(-t) and below.

#### Fourier Transform

The Fourier transform of x(t) is given by

(1)

#### Fourier Transform of Conjugate

The Fourier transform of , the conjugate of x(t), is given by

(2)

A double-conjugate is applied to (2)

(3)

and the inner conjugate is simplified as

(4)

A variable substitution is made for the frequency ,

(5)

such that (4) is written as

(6)

Using (1) the integral in (6) is simplified to

(7)

such that

(8)

#### Fourier Transform of Time Reversal

The Fourier transform of x(-t), which is time reversed x(t), is given by

(9)

Substitute such that (9) is written as

(10)

where

(11)

Swapping the integration limits in (10) cancels the negation so the integral is written as

(12)

Substitute for the frequency f,

(13)

such that (12) is written as

(14)

Using (1) the Fourier transform of x(-t) is written as

(15)

which is simplified as

(16)

The blog on negative frequency showed that positive frequencies rotate in a counter-clockwise direction around the unit circle. My interpretation of (16) is that reversing the time, x(-t), also reverses the direction rotation around the unit circle for positive frequencies, which is why the frequencies in X(-f) are reversed as well.

#### Fourier Transform of Conjugate and Time Reversal

The Fourier transform of , the conjugated and time reversal of x(t), is given by

(17)

A double conjugate is applied over (17),

(18)

and evaluating the inner conjugate leads to

(19)

A variable substitution is made for the time,

(20)

where

(21)

such that (19) is written as

(22)

Swapping the integration limits cancels the negation such that (22) is written as

(23)

The integral in (23) is evaluated as

(24)

such that

(25)

#### Conclusion

Don’t forget these posts on DSP Math!

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

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