Testwiki:Reference desk/Archives/Mathematics/2016 February 7
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February 7
Semi-hereditary rings
Is there a ring R that is left semi-hereditary but not right semi-hereditary? Of course, the opposite ring Rop will then be right semi-hereditary but not left semi-hereditary. Unlike for (semi-)perfect rings and (semi-)firs, where the semi version is left-right symmetric and the non-semi version is asymmetric, both hereditariness and semi-hereditariness are asymmetric. GeoffreyT2000 (talk) 00:24, 7 February 2016 (UTC)
Discrete Fourier Transform
By convention, when we take a DFT of a series, we get a series-sized list of numbers back. These numbers describe the Frequency domain of that series. My question is: what is the exact relation of each number to the original signal? Let's say we take a DFT of 1024 samples from an audio recording with a sample rate of 44100 Hz. We get back a list of 1024 numbers. The first number (or last depending on how you order it I guess, but by convention usually the first) will represent the "constant" signal of the time series, correct? The last number will represent a signal oscillating fast enough to go through a full sine wave 512 times over the course of our 1024 samples (alternating between full positive and full negative every sample), right? This corresponds to a frequency of 22050 Hz? So what does e.g. the 384th number represent?
tl;dr: I wanna tie the results of an FFT of an audio file to specific frequencies in Hertz. How do? 97.93.100.146 (talk) 21:58, 7 February 2016 (UTC)
- Hi, reporting back on some of my own digging to help build the record. My first point of confusion is I've been working with a full FFT instead of a real FFT when working with Real data. The "extra" 512 coefficients are identical to the first 512 when working with real data! So, with a RFFT of the same data from the earlier example, the 512th (e.g. last) member is 22050 hz. 97.93.100.146 (talk) 22:26, 7 February 2016 (UTC)
- The resulting values are evenly spaced in frequency domain. So if you have N samples and a sample rate of S, then the k-th resulting value corresponds to frequency . As you already noted, k > N / 2 just repeats for real signals, so the interesting information covers frequencies S / N to S / 2. Dragons flight (talk) 11:41, 8 February 2016 (UTC)
- Thank you! For some reason I was just assuming it'd be a nonlinear relation. So e.g. in a FFT of 1 second of 44.1khz audio you'd have 22051 numbers; a static offset/whatever you wanna call it, and one for each integral frequency between 1 and 22050. 97.93.100.146 (talk) 19:34, 8 February 2016 (UTC)
- The resulting values are evenly spaced in frequency domain. So if you have N samples and a sample rate of S, then the k-th resulting value corresponds to frequency . As you already noted, k > N / 2 just repeats for real signals, so the interesting information covers frequencies S / N to S / 2. Dragons flight (talk) 11:41, 8 February 2016 (UTC)