Basic Discrete Time Signals

The unit step sequence is defined as 1 for zero and positive values of the integer n. This sequence can be graphically displayed using a set of eight sample points, showing a step function starting from n=0 and remaining constant thereafter.

The unit impulse or sample sequence is mathematically expressed as zero for all n values except at n=0, where it is one. The unit impulse sequence, denoted by δ(n), is the first difference of the unit step sequence, while the unit step sequence u(n) is the cumulative sum of the unit impulse sequence. This relationship can be visually demonstrated through a graph, highlighting how the unit impulse can effectively sample the signal value at n=0.

The unit ramp sequence exhibits a linear increase in value with the increase in the sample number. For instance, a sequence of 12 samples on a unit ramp will show a linear increase in amplitude with each sample number, represented graphically as a straight line.

A sinusoidal sequence is defined by its amplitude and phase parameters. This sequence can be represented as,

where A is the amplitude, ω is the angular frequency, and Φ is the phase.

The exponential sequence is defined using complex numbers, with exponentially decaying and increasing sequences represented on a graph. An exponentially decaying sequence can be written as,

while an exponentially increasing sequence is expressed as,

where A is the initial amplitude and α is a positive constant. These sequences are fundamental in analyzing various signal-processing applications due to their unique properties and behaviors.