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Unformatted text preview: DiscreteTime Fourier Series Do not distribute these notes 169 Prof. R.T. M’Closkey, UCLA Discretetime Signals A continuoustime signal produces a discretetime signal through the process of sampling . The most common form of sampling takes a snapshot of the continuoustime signal at equally spaced points in time. The sample period , denoted t s , is the time interval separating adjacent samples. The discretetime signal becomes a repre sentation of the continuoustime signal. If the continuoustime signal is u ( t ) then the discretetime signal is the set { u p } where u p = u ( pt s ) , p ∈ Z ( Z = set of integers). One question that must be asked is “when is the discretetime signal a faithful representation of the continuoustime signal?" ï 2 ï 1.5 ï 1 ï 0.5 0.5 1 1.5 2 ï 2 ï 1.5 ï 1 ï 0.5 0.5 1 1.5 2 second volts ï 2 ï 1.5 ï 1 ï 0.5 0.5 1 1.5 2 ï 2 ï 1.5 ï 1 ï 0.5 0.5 1 1.5 2 second volts continuoustime signal discretetime signal a clock triggers the ADC to take a “snapshot” of the continuoustime signal these points are stored in memory as a representation of the continuoustime signal t s = 0 . 2 We will answer this question in the context of sampling periodic continuoustime signals. Do not distribute these notes 170 Prof. R.T. M’Closkey, UCLA Discretetime Sinusoids A discretetime sinusoid is generated when a continuoustime sinusoid is sampled. For example, given frequency ω (which we will assume to be positive without loss of generality) the continuoustime (complex) sinusoid is defined as u ( t ) = e jωt , t ∈ (∞ , ∞ ) . If this signal is sampled with a sample period t s , the following discretetime sinusoid is generated u p := u ( pt s ) = e jωpt s , p ∈ Z ( Z =set of integers ) . (69) We introduce the notation ω s = 2 π t s to denote the sampling frequency or sampling rate in rad/s. If ω > ω s / 2 , though, we will show that it is possible to generate the same sequence { u p } by sampling a continuoustime sinusoid of lower frequency in the interval [0 , ω s / 2] . For example, rewrite (69) as u p = e j 2 π ω ωs p , p ∈ Z , and suppse 1 2 < ω/ω s ≤ 1 , i.e. ω ∈ ( 1 2 ω s , ω s ] . Define a new frequency, denoted ¯ ω , as follows: ¯ ω := ω s ω → ¯ ω ω s = 1 ω ω s . Note that ¯ ω ∈ [0 , ω s / 2) so it is a lower frequency. Substituting this definition into the expression for u p yields, u p = e j 2 π ( 1 ¯ ω ωs ) p = e j 2 πp e j 2 π ¯ ω ωs p = e j 2 π ¯ ω ωs p p ∈ Z . In other words, { u p } can be generated by sampling a sinusoid with frequency ¯ ω . As another example, suppose ω ∈ [ ω s , 3 2 ω s ] and define ¯ ω := ω ω s which implies ¯ ω ∈ [0 , 1 2 ω s ] . Note that ¯ ω ω s = ω ω s 1 , so substituting this relation into (69) yields u p = e j 2 π ( 1+ ¯ ω ωs ) p = e j 2 πp e j 2 π ¯ ω ωs p = e j 2 π ¯ ω ωs p p ∈ Z , Do not distribute these notes 171 Prof. R.T. M’Closkey, UCLA so once again the discretetime sinusoid can be generated by sampling a continuoustime sinusoid with frequency...
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This note was uploaded on 08/08/2011 for the course MAE 107 taught by Professor Tsao during the Spring '06 term at UCLA.
 Spring '06
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