We have seen, previously, these four important facts:
--the output of infinite-length LTI systems can be found via linear convolution--linear convolution can be found through circular convolution, through zero-padding
--there is an equivalence between circular convolution in the signals' time domain and multiplication in their frequency domain--the FFT is an algorithm that can quickly compute a DFT
Perhaps at the time it may have seemed that these findings were unrelated to each other. But as we now string them together, you can see the incredibly significant consequence: the output of LTI systems can be computed very efficiently. This truth has supported the incredible advances in signal processing over the past fifty years.
Putting it all together...
Suppose we have an LTI system with an impulse response of $h[n]$ and an input of $x[n]$. We would like to find the output, $y[n]$. We could find this output via linear (infinite length) convolution. If the length of the impulse response is $N_h$ and that of the input signal is $N_x$, then about $N_h N_x$ operations would be required to compute this convolution. It would also be possible to compute the output through circular convolution, by zero-padding each signal to be of length $N_h+N_x-1$, and then circularly convolving the zero-padded signals. This would not save any computational steps, but it is a significant insight because the circular convolution could be performed by using DFTs: simply take the DFT of each zero-padded signal, multiply the two DFTs together, then take the inverse DFT of the result. Again, this does not at first seem to save any computational steps, and in fact seems to add even more, except that we have seen that the FFT is able to perform DFTs in about $N\log N$ operations. What that means is that if we zero-pad $x[n]$ and $h[n]$, then take the DFT of each (using the FFT algorithm), multiply these two together, then take the inverse DFT of the result, we can find the system output in about $2(N_h+N_x-1)\log_2 (N_h+N_x-1)$ operations. As signal lengths increase, this can end up being huge computational savings over the $N_h N_x$ operations it would take to find the output through a convolution sum. Below is a graphical depiction of the steps to quickly find the output, with the help of the FFT:
Questions & Answers
I'm interested in biological psychology and cognitive psychology
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
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