<< Chapter < Page Chapter >> Page >

The savings in arithmetic can be considerable when implementing convolution or performing FIR digital filtering. However, there are two penalties. The use of blocks introduces a delay of one block length. None of the first block of output can be calculated until all of the first block of input is available. This is not a problem for “off line" or “batch" processing but can be serious for real-time processing. The second penalty is the memory required to store and process the blocks. The continuing reduction of memory cost often removes this problem.

The efficiency in terms of number of arithmetic operations per output point increases for large blocks because of the M log ( M ) requirements of the FFT. However, the blocks become very large ( L > > M ), much of the input block will be the appended zeros and efficiency is lost. For any particular application, taking the particular filter and FFT algorithm being used and the particular hardware being used, a plot of efficiency vs. block length, L should be made and L chosen to maximize efficiency given any other constraints that are applicable.

Usually, the block convolutions are done by the FFT, but they could be done by any efficient, finite length method. One could use “rectangular transforms" or “number-theoretic transforms". A generalization of this method is presented later in the notes.

Fast convolution by overlap-save

An alternative approach to the Overlap-Add can be developed by starting with segmenting the output rather than the input. If one considers the calculation of a block of output, it is seen that not only the corresponding input block is needed, but part of the preceding input block also needed. Indeed, one can show that a length M + L - 1 segment of the input is needed for each output block. So, one saves the last part of the preceding block and concatenates it with the current input block, then convolves that with h ( n ) to calculate the current output

Block processing, a generalization of overlap methods

Convolution is intimately related to the DFT. It was shown in The DFT as Convolution or Filtering that a prime length DFT could be converted to cyclic convolution. It has been long known [link] that convolution can be calculated by multiplying the DFTs of signals.

An important question is what is the fastest method for calculating digital convolution. There are several methods that eachhave some advantage. The earliest method for fast convolution was the use of sectioning with overlap-add or overlap-save and the FFT [link] , [link] , [link] . In most cases the convolution is of real data and, therefore, real-data FFTs should be used. That approach is stillprobably the fastest method for longer convolution on a general purpose computer or microprocessor. The shorter convolutions shouldsimply be calculated directly.

Introduction

The partitioning of long or infinite strings of data into shorter sections or blocks has been used to allow application of the FFT to realizeon-going or continuous convolution [link] , [link] . This section develops the idea of block processing and shows that it is a generalizationof the overlap-add and overlap-save methods [link] , [link] . They further generalize the idea to a multidimensional formulation ofconvolution [link] , [link] . Moving in the opposite direction, it is shown that, rather than partitioning a string of scalars into blocks andthen into blocks of blocks, one can partition a scalar number into blocks of bits and then include the operation of multiplication in the signalprocessing formulation. This is called distributed arithmetic [link] and, since it describes operations at the bit level, is completely general. These notes try to present a coherent development of theseideas.

Questions & Answers

I'm interested in biological psychology and cognitive psychology
Tanya Reply
what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Fast fourier transforms (6x9 version). OpenStax CNX. Aug 24, 2009 Download for free at http://cnx.org/content/col10683/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fast fourier transforms (6x9 version)' conversation and receive update notifications?

Ask