Changes between Version 52 and Version 53 of Other/Summer/2024/signal
- Timestamp:
- Aug 6, 2024, 7:08:17 PM (4 months ago)
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Other/Summer/2024/signal
v52 v53 45 45 Because every signal is composed of sine waves of varying frequency and phase, we can actually represent the signals in terms of the frequencies and phases of those sine waves. This is called the Frequency Domain and is an equivalent representation for any signal. Below and left is a square wave in the time domain and to the right is the same signal in the frequency domain. 46 46 47 [[Image(Screen Shot 2024-08-06 at 3.01.41 PM.png, 300px)]]47 [[Image(Screen Shot 2024-08-06 at 3.01.41 PM.png, 400px)]] 48 48 49 49 The conversion from time domain to the frequency domain occurs by multiplying each possible frequency sine wave with the signal and integrating over 1 period. This gives the magnitude of that frequency in the signal. In reality, an algorithm call Fast Fourier Transform (FFT) is used to do this multiplication and integration much quicker. FFT is a divide and conquer algorithm that makes use of complex number properties. 50 50 51 [[Image(Screen Shot 2024-08-06 at 3.01.51 PM.png, 300px)]]51 [[Image(Screen Shot 2024-08-06 at 3.01.51 PM.png, 400px)]] 52 52 53 53 **Complex Signals** … … 55 55 Speaking of complex number properties, signals can also be represented as complex numbers. While we may think of signals as up and down waves, shown left. In reality, signals can be complex 3 dimensional patterns, shown right. 56 56 57 [[Image( )]]58 [[Image( )]]57 [[Image(Screen Shot 2024-08-06 at 2.31.04 PM.png)]] 58 [[Image(Screen Shot 2024-08-06 at 2.31.18 PM.png)]] 59 59 60 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.
