1. Convergence of the Fourier Series Used in Example 2.3 (and why analysis of the harmonics is important in communication systems)

2. Let’s see how the stairstep signal of Example 2.3 converges as we add more harmonics to its Fourier Series summation -0.5 0.5 1 1.5 2 2.5 -2-1.5-0.50.511.5 22.533.544.5 55.5 66.5 77.5 8 volts sec

3. -0.5 0.5 1 1.5 2 2.5 -2-1.5-0.50.511.5 22.533.544.5 55.5 66.5 77.5 8 Summation of dc term plus first five harmonics volts sec

4. Summation of dc term plus first ten harmonics -0.5 0.5 1 1.5 2 2.5 -2-1.5-0.50.511.5 22.533.544.5 55.5 66.5 77.5 8 volts sec

5. Summation of dc term plus first fifteen harmonics -0.5 0.5 1 1.5 2 2.5 -2-1.5-0.50.511.5 22.533.544.5 55.5 66.5 77.5 8 volts sec

6. -0.5 0.5 1 1.5 2 2.5 -2-1.5-0.50.511.5 22.533.544.5 55.5 66.5 77.5 8 Summation of dc term plus first twenty- five harmonics volts sec

7. -0.5 0.5 1 1.5 2 2.5 -2-1.5-0.50.511.5 22.533.544.5 55.5 66.5 77.5 8 Summation of dc term plus all harmonics volts sec

8. Let’s pass the stairstep function from Example 2.3 through channels with various bandwidths and evaluate the distortion in the received signal. 1 2 01 2 3 4 5 6 7 8 sec volts The importance of analyzing harmonics in communication systems

9. 1 2 01 2 3 4 5 6 7 8 sec volts Channel with 1 Hz bandwidth Channel with 1 Hz bandwidth passes the first five harmonics 1 2 01 2 3 4 5 6 7 8 sec volts

10. 1 2 01 2 3 4 5 6 7 8 sec volts Channel with 2 Hz bandwidth Channel with 2 Hz bandwidth passes the first ten harmonics 1 2 01 2 3 4 5 6 7 8 sec volts

11. 1 2 01 2 3 4 5 6 7 8 sec volts Channel with 3 Hz bandwidth Channel with 3 Hz bandwidth passes the first fifteen harmonics 1 2 01 2 3 4 5 6 7 8 sec volts

12. 1 2 01 2 3 4 5 6 7 8 sec volts Channel with 5 Hz bandwidth Channel with 5 Hz bandwidth passes the first twenty-five harmonics 1 2 01 2 3 4 5 6 7 8 sec volts

13. 1 2 01 2 3 4 5 6 7 8 sec volts Channel with infinite bandwidth passes all harmonics 1 2 01 2 3 4 5 6 7 8 sec volts Channel with infinite bandwidth For the stairstep function, we now see how channel bandwidth and accuracy of the received signal are related.