# Wave Walker DSP

## Trending

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

Deriving the Ideal Low Pass Filter (LPF)
October 1, 2022

#### Introduction

This blog derives the ideal low pass filter (LPF) step by step using simple math.

More DSP blogs!

#### Frequency Domain Ideal Low Pass Filter

An ideal low pass filter (LPF) allows signals below the cutoff frequency to pass through unmodified with a linear gain of 1 and no phase change, while simultaneously completely rejecting all frequencies above the cutoff frequency with a linear gain of 0.

Mathematically, the ideal LPF is defined in the frequency domain by [oppenheim1999, p.43]

(1)

The frequency response for the ideal LPF in (1) for is given in Figure 1.

#### Time Domain Ideal Low Pass Filter

The ideal LPF in the time domain is derived by taking the inverse discrete-time Fourier transform (IDTFT) of (1),

(2)

The IDTFT is defined as

(3)

Substituting (1) into (3),

(4)

The integral in (4) is simplified via Euler’s formula as

(5)

The cutoff frequency is the frequency in radians, defined as

(6)

where is the frequency in Hz normalized by the sampling rate ,

(7)

The impulse response h[n] can therefore be written as

(8)

Equation (8) can then be transformed into a sinc function where

(9)

The impulse response h[n] can therefore be written as

(10)

The impulse response from (10) is given in Figure 2.

#### Conclusion

The ideal low pass filter is a finite-length rectangle in the frequency domain but an infinitely long sinc in the time domain. Selecting the cutoff frequency determines which frequencies will be passed by the filter and which will be attenuated.

More DSP blogs!

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.