Chapter 01 Discrete Sequences and Systems

1.1 DISCRETE SEQUENCES AND THEIR NOTATION

Think of a continuous sinewave with a peak amplitude of $1$ at a frequency $f_o$ described by the equation

$\tag{1-1} x(t) = \sin 2 \pi f_o t$

Those sample values can be represented collectively, and concisely, by the discrete-time expression

$\tag{1-3} x(n) = \sin 2 \pi f_o n t_s$

If we assume that the proportionality constant is one, we can express the power of a sequence in the time or frequency domains as

$\tag{1-8} x_{pwr}(n) = |x(n)|^2$

or

$\tag{1-8'} X_{pwr}(m) = |X(m)|^2$

$a(n) = b(n) + c(n)$

$a(n) = b(n) - c(n)$

$\begin{split} a(n) &= \sum_{k=n}^{n+3} b(k) \\ &= b(n) + b(n+1) + b(n+2) + b(n+3) \end{split}$

$a(n) = b(n)c(n) = b(n) \cdot c(n)$

If we know the unit impulse response of an LTI system, we can calculate everything there is to know about the system; that is, the system’s unit impulse response completely characterizes the system.
By “unit impulse response” we mean the system’s time-domain output sequence when the input is a single unity-valued sample (unit impulse) preceded and followed by zero-valued samples as shown in Figure 1–11(b).
The concepts in the two following sentences are among the most important principles in all of digital signal processing!
- Knowing the (unit) impulse response of an LTI system, we can determine the system’s output sequence for any input sequence because the output is equal to the convolution of the input sequence and the system’s impulse response.
- Given an LTI system’s time-domain impulse response, we can find the system’s frequency response by taking the Fourier transform in the form of a discrete Fourier transform of that impulse response
In the world of DSP, an impulse sequence called a unit impulse takes the form

$\tag{1-26} x(n) = \cdots,0,0,0,0,0,A,0,0,0,0,0,\cdots$

The leading 0 must be long enough, so the system is outputing 0.
The trailing 0 must also be long enough, so it's longer than the transient response.