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# Introduction

Before looking at this module, hopefully you have an idea of what a signal is and what basic classifications and properties a signal can have. In review, a signal is a function defined with respect to an independent variable. This variable is often time but could represent any number of things. Mathematically, discrete time analog signals have discrete independent variables and continuous dependent variables. This module will describe some useful discrete time analog signals.

# Sinusoids

One of the most important elemental signal that you will deal with is the real-valued sinusoid. In its discrete-time form, we write the general expression as

$A⁢cosω⁢n+φ A ω n φ$
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where $A A$ is the amplitude, $ω ω$ is the frequency, and $φ φ$ is the phase. Because n only takes integer values, the resulting function is only periodic if $2⁢πω 2 ω$ is a rational number.
Discrete-Time Cosine Signal Figure 1: A discrete-time cosine signal is plotted as a stem plot. Note that the equation representation for a discrete time sinusoid waveform is not unique.

# Complex Exponentials

As important as the general sinusoid, the complex exponential function will become a critical part of your study of signals and systems. Its general discrete form is written as

$A⁢es⁢n A s n$
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where $s=σ+j⁢ω s σ j ω$ , is a complex number in terms of $σσ$, the attenuation constant, and $ωω$ the angular frequency.

The discrete time complex exponentials have the following property.

$ej⁢ω⁢n=ej⁢(ω+2⁢π)⁢n j ω n j ω 2 n$
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Given this property, if we have a complex exponential with frequency $ω+2⁢π ω 2$, then this signal "aliases" to a complex exponential with frequency $ωω$, implying that the equation representations of discrete complex exponentials are not unique.

# Unit Impulses

The second-most important discrete-time signal is the unit sample, which is defined as

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Unit Sample

More detail is provided in the section on the discrete time impulse function. For now, it suffices to say that this signal is crucially important in the study of discrete signals, as it allows the sifting property to be used in signal representation and signal decomposition.

# Unit Step

Another very basic signal is the unit-step function defined as

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