 case, you can instead figure out the outputy(t), given the input x(t), by using the relationship Y(f)=H(f)x(f) (3.16) where Y(f), X(f), and H(f) are the Fourier transforms of y(t),x(t), and (t), respectively. 3.3.5 Some Special Linear Time Invariant (LTI) Systems Earlier, I talked about some special signals. Now, I'll talk about some special systems. I'll use them a lot throughout this book, SO it's rather important you know what these systems are. First, there's a linear time invariant system called a low-pass filter, or LPF for short. This refers to a system where H(f) is as shown in Figure 3.12. In other words, it's a system where the H(f) is made mostly of frequencies less than fm. Take a look at Figure 3.13. Here, you can see the signal x(t) (i.e., X(f)) coming into the system. The system has an H(f) that makes it a low-pass filter. What comes out of the system is System characterized by Figure 3.12 LTI system corresponding to LPF A Review of Some Important Math, Stats, and Systems Y(f) = H(f)X(f), and this is also drawn in Figure 3.13. You can see the output corre- sponds simply to the low-frequency components of the input. Hence, the name low-pass filter (LPF), because only the low frequencies pass through to the output. Next, there's a linear time invariant system called a bandpass filter (BPF for short). You can understand the BPF by studying Figure 3.14. It has an H(f) as shown in the figure; in words, the H(f) is not 0 only around some frequency fc. Now, look at what happens when an input (t) (X(f)) enters into the BPF. In Figure 3.14, the output is made up of just the frequencies around fc. That is, the BPF only allows frequencies around some f to get to the output. LTI System Figure 3.13 What happens in an LPF LTI System Figure 3.14 What happens in a BPF Chapter Three LTI System Inverse LTI System h-1(t) Figure 3.15 Describing inverse LTI system Finally, we've got an inverse system. For a given LTI system with impulse response h(t) (H(f)), the inverse system is defined as follows: (1) in pictures, t