PHYS 352. FFT Convolution. More Advanced Digital Signal Processing Techniques

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Derrick Simmons
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1 PHYS 352 More Advanced Digital Signal Processing Techniques FFT Convolution take a chunk of your signal (say N=128 samples) apply FFT to it multiply the frequency domain signal by your desired transfer function inverse FFT back to time domain that s your output signal sounds easy? and it works!

2 FFT Convolution versus FIR Filter if you can acquire all of your data in memory (disk), and only need to do offline processing, you can certainly use FFT convolution to achieve ideal low-pass filter response how ideal is ideal? how sharp frequency response can be acheived? Δf = 1/T, where T is the total length in time of the N samples T = NΔt, where Δt is the time interval between samples f s = 1/Δt, where f s is the sampling frequency Δf = f s / N can be faster in computation than the standard FIR filter, especially for long kernels FFT Convolution in Real Time? however, in real-time applications, it may not be possible for FFT convolution, of the desired quality, to keep up take a chunk of data FFT it note: smaller chunks have less Δf resolving ability apply the transfer function inverse FFT it produce your output must be ready for the next chunk of data FFT convolution filter quality is ultimately limited by computation time just as for FIR filters (where the sharpness of the roll-off comes at the expense of kernel length)

3 Output of FFT Convolution the output extends to later times than the input why? it s because you are multiplying by a transfer function in frequency domain this is mathematically equivalent to time domain convolution output length: N+M 1 due to time domain convolution e.g. last x value [128] but y[130]=...+h[2]x[130-2]+h[1]x[130-1]+... Circular Convolution an Error! example: 128 samples in the input FFT give 64 complex-valued frequency components you manipulate those 64 frequency components (multiplying with a 64-component complex-valued transfer function) that s as though you were convolving with a 128- point impulse response in frequency domain you have 64 complex-valued frequency components waiting for inverse FFT you can only generate 128 samples in your output but, time domain convolution would insist that the output length is 255 samples those output samples that get pushed off the end (to the right) roll back into the beginning (at the left) introduces what s known as circular convolution error

4 Overlap-Add overlap-add applies in time-domain filtering and hence must be necessary in FFT convolution filtering (because, they are the same mathematically) Complication #1 requires padding with zeroes say you want to use a 1024-point FFT contains real-valued c 0 and c 512, complex-valued c 1 c 511, or 1024 real values contains frequencies c 1 c 511 which are not independent values inverse FFT would generate 1024 samples in the output for example, you could use a kernel with 600 non-zero points and 424 padded zeros at the end you could use an input with 425 values and 599 padded zeros at the end...it s going to be N+M 1 the end result is that a 1024-point input is fed to the FFT, you always have 1024 points in frequency domain, you multiply bin-by-bin with a 1024 point transfer function, and your output is ready for inverse FFT, and has 1024 samples

5 Complication #2 negative frequencies don t forget about those! say you want to perform zero-phase, unity gain, low-pass, from 0-5 rad/s (ω) your transfer function should process the negative frequencies unless you want trouble from -5 to 5: f(t) = 5/π sinc(5t) from 0 to 5: f(t) = 5/(2π) sinc(5t/2) e jt5/2 if you forget to process the negative frequencies, your transfer function is equivalent to time domain convolution with a complex impulse response!?! that s very bad! Complication #3 phase zero phase results in an impulse response that is symmetrical about t=0 non-causal! your output signal will have values at earlier times than your input signal Fourier transform pairs: f(t-t 0 ) time delayed impulse response (to preserve causality) implies linear phase so it s best not to ignore phase when you process in frequency domain

6 FFT Convolution Summary faster, and much faster for large kernels even more ideal filter performance can be achieved but more complicated complex-valued FFT with negative frequencies complex-valued multiplication in frequency domain don t forget about phase! pad your input and impulse response with extra zeroes overlap-add to stitch together the time domain output in contrast FIR filter: easy to implement (addition/multiplication of real numbers) FFT Signal Processing - Deconvolution deconvolution is certainly a place for FFT signal processing... your signal goes through a physical system (or measurement system) and that process alters the signal the system has convolved it s response function with your signal if you want to get back your original signal you must do deconvolution deconvolution in time domain is fraught with peril deconvolution in frequency domain is trivial in comparison

Hideo Okawara s Mixed Signal Lecture Series. DSP-Based Testing Fundamentals 14 FIR Filter

Hideo Okawara s Mixed Signal Lecture Series DSP-Based Testing Fundamentals 14 FIR Filter Verigy Japan June 2009 Preface to the Series ADC and DAC are the most typical mixed signal devices. In mixed signal