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The image you uploaded contains a question about the square wave function, Fourier series coefficients, and Fourier sum approximations. The question consists of multiple parts:

### a) Function for Square Wave:
- You're asked to write a function that evaluates the square wave at any value of \( t \), given that the signal is periodic with a period \( T_0 = 1 \). The square wave alternates between \( 1 \) and \( 0 \) for half of the period each.

### b) Fourier Series Coefficients:
- You need to compute the Fourier series coefficients \( a_k \) for the square wave \( g(t) \), where \( k \) ranges from \( -20 \) to \( 20 \). You then need to make a stem plot of the magnitudes of these coefficients.

### c) Fourier Sum Approximation:
- You're asked to generate Fourier sum approximations for \( N = 3 \), \( N = 10 \), and \( N = 20 \). This part is related to how the Fourier series approximates the square wave.

### d) Synthesis Errors:
- Finally, you're asked to plot the synthesis errors \( |g_N(t) - g(t)| \) for \( N = 3 \), \( N = 10 \), and \( N = 20 \), where \( g_N(t) \) is the partial Fourier sum.

Would you like me to walk you through solving these steps or assist with any specific part of the problem?

Now consider the square wave function, which is defined over one period as g(t) = {1, for 0 ≤ t < 1/2 T0; 0, for 1/2 T0 ≤ t < T0}. The signal is periodic, so the square wave should repeat every period. Note that g(t) is not continuous! a) Let T0 = 1, and write a function that evaluates the square wave at any value of t. Plot three seconds of the signal. b) Compute the Fourier series coefficients {ak} for k = -20, ..., 20 of g(t). Make a stem plot of the magnitudes of these coefficients. c) Generate and plot the partial Fourier sum approximations for N = 3, N = 10, and N = 20. d) Make plots of the synthesis errors |gN(t) - g(t)| for N = 3, 10, and 20, where gN(t) are the partial Fourier sums.

Learn how to evaluate the square wave function, compute its Fourier series coefficients, and analyze synthesis errors. Includes plots for partial Fourier sums at N = 3, 10, and 20, with a focus on periodic signals and signal processing concepts. Suitable for undergraduate engineering and applied mathematics students.

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