Changes between Version 70 and Version 71 of Other/Summer/2024/signal
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- Aug 6, 2024, 9:51:31 PM (4 months ago)
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Other/Summer/2024/signal
v70 v71 41 41 You may notice that this looks strikingly similar to some sort of sine wave and you would be 100% correct. Every signal can be thought of as a combination of sine waves. Each of theses sine waves has its own frequency and phase, meaning how fast it goes up and down and where the sine wave is centered, respectively. 42 42 43 ---- 44 43 45 **Frequency Domain** 44 46 … … 58 60 [[Image(Screen Shot 2024-08-06 at 2.31.18 PM.png, 400px)]] 59 61 60 The axis going into the page is the time and the vertical / horizontal axis are the phase and amplitude respectively. There are many other ways to think about these axis as well including the coefficients of sine and cosine waves and most notably real and imaginary numbers. This representation gives rise to a new way to represent the signal algebraically as well. S(t) = I(t) + Q(t) i, where S(t) is the signal, I(t) is the real component, Q(t) is the imaginary component and i is the imaginary number sqrt(-1). This imaginary number representation is advantageous in making the math in later stages much simpler.62 The axis going into the page is the time and the vertical / horizontal axis are the phase and amplitude respectively. There are many other ways to think about these axis as well including the coefficients of sine and cosine waves and most notably real and imaginary numbers. This representation gives rise to a new way to represent the signal algebraically as well. S(t) = I(t) + Q(t)''i'', where S(t) is the signal, I(t) is the real component, Q(t) is the imaginary component and i is the imaginary number sqrt(-1). This imaginary number representation is advantageous in making the math in later stages much simpler. 61 63 62 64 [[Image(Screen Shot 2024-08-06 at 3.02.06 PM.png, 300px)]] … … 68 70 In order to send signals as higher frequencies, we use a technique call modulation. We take the real part, I(t), of a signal and multiply it with a sine wave of the desired high frequency and the imaginary part, Q(t) with a cosine wave of that same frequency. 69 71 70 S(t) = I(t) + Q(t) i-> I(t)*sin(wt) + Q(t)*cos(wt)72 S(t) = I(t) + Q(t)''i'' -> I(t)*sin(wt) + Q(t)*cos(wt) 71 73 72 74 Note how the resulting signal is always real, even if the original signal had an imaginary component. It is important to remember that even though complex/imaginary signals are useful for various reasons, they don't actually exist in the real world because... well, they are imaginary! … … 96 98 In order to determine the bits from a signal we first have to get our IQ data from the signal. This is done by multiplying by sine and cosine waves of the carrier frequency. 97 99 98 S(t) = I(t) + Q(t) i100 S(t) = I(t) + Q(t)''i'' 99 101 100 S(t)*sin(wt) -> low pass filter -> I(t)\\ 101 102 S(t)*sin(wt) -> low pass filter -> I(t) 102 103 S(t)*cos(wt) -> low pass filter -> Q(t) 103 104
