DSP Algorithms for RF Systems

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List of Important Math for DSP

The following post is a list of math for DSP I have found to be useful over my career in DSP. This post will be updated periodically and will serve as a reference for upcoming material.

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DSP Wisdom

I’ve compiled a list of DSP wisdom I have learned and used over my career. The list is a collection of aphorisms in the vein of Nassim Taleb’s Bed of Procrustes and Epictetus’ Handbook. This blog post will be updated over time and used as a reference in other posts.

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A graphical representation of the system dependency list used in the product delivery plan.

Reverse Engineer Your Product Delivery Plan

You have been assigned a product delivery on a too-short of a deadline.

Let’s make it harder and say you just got promoted to the project lead and have a couple of junior engineers working for you. You have to build a schedule, allocate people to different tasks, report your progress to your leadership and your customer and get everything synchronized so it comes together successfully at the end.

Let’s add more.

The success of the project has implications to your career and reputation, as well as your organization. Your success may have implications to the profitability or success of your company. All eyes are on you.

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Figure 1: The unit circle with an example complex vector e(j pi/3).

The Unit Circle: The Field of Play for the Game of DSP

Thinking about complex sinusoids, \sqrt{-1} = j, the unit circle, phase, negative frequency and using \pi instead of 180^\circ can all be confusing when the material is first encountered. Have you asked yourself the following questions? I certainly have:

  • Why do we need the complex plane?
  • Why can’t we just use degrees instead of radians?
  • Why is the unit circle useful?

Unfortunately many of the mathematical tools encountered in the first and second years of an undergraduate EE degree are abstract and their value may not seem practical until graduate school or your first job as an engineer. The hope is that the following post will connect these concepts into a real world situation using the face of a clock in an attempt to build some intuition.

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Free DSP Code!

Some posts have code included to help demonstrate the concepts. You can check them out under the post with code tag. Enjoy!

Figure 4: The time-domain plot of e(j2 pi n/8) for n=0 to n=31.

Phase, Frequency and Negative Frequency in Complex Sinusoids

A prior post described how sinusoids came to be used in DSP and RF due to physical properties of the universe. A follow up gave some mathematical rules about complex sinusoids on how sine, cosine and e^{j\phi} all related to one another in terms of Euler’s triangle. This blog will continue by including time in complex sinusoids, describe how phase and frequency relate to one another, how a negative frequency arises and demonstrates the concepts using the unit circle. It may be useful to review some complex math before jumping into this post.

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Figure 4: The PFB half band filter structure after folding the even-symmetric filter weights hA[n] and ignoring the zero weights in hB[n].

Folding a Polyphase Half Band Filter: Lemon Squeezer Style

Folding a filter is a way to improve efficiency, and folding the weights for a polyphase filterbank improves the efficiency beyond that. The zero weights in the half band filter make for a great application of the folding and polyphase filterbank structures.

You know after you cut a lemon and try to get the juice out by hand you get to the point where you need a lemon squeezer to get every last drop? This blog is going to squeeze out every little bit of efficiency in the polyphase half band filter.

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Figure 1: The weights of the partitioned half band filter hA[n] and hB[n].

Polyphase Half Band Filter for Decimation by 2

The last post on half band filters (HBF) referenced the use of a polyphase filter bank structure with a half band filter of length N can be reduced to N/8 multiplies per input sample. This is a huge efficiency gain and why they are used in large sample rate change [harris2021, p.234]. The polyphase filter bank will be used to efficiently implement a decimation by 2 within the HBF with additional savings coming from folding the filter weights. A polyphase filterbank is characterized by multiple branches which represent multiple phases of the signal (the prefix poly- meaning “many”.)

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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
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