discrete_time_fourier

discrete_time_fourier - Discrete-Time Fourier Series Do not...

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Discrete-Time Fourier Series Do not distribute these notes 169 Prof. R.T. M’Closkey, UCLA
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Discrete-time Signals A continuous-time signal produces a discrete-time signal through the process of sampling . The most common form of sampling takes a snapshot of the continuous-time signal at equally spaced points in time. The sample period , denoted t s , is the time interval separating adjacent samples. The discrete-time signal becomes a repre- sentation of the continuous-time signal. If the continuous-time signal is u ( t ) then the discrete-time 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 discrete-time signal a faithful representation of the continuous-time signal?" ï 2 ï 1.5 ï 1 ï 0.5 0 0.5 1 1.5 2 ï 2 ï 1.5 ï 1 ï 0.5 0 0.5 1 1.5 2 second volts ï 2 ï 1.5 ï 1 ï 0.5 0 0.5 1 1.5 2 ï 2 ï 1.5 ï 1 ï 0.5 0 0.5 1 1.5 2 second volts continuous-time signal discrete-time signal a clock triggers the ADC to take a “snapshot” of the continuous-time signal these points are stored in memory as a representation of the continuous-time signal t s = 0 . 2 We will answer this question in the context of sampling periodic continuous-time signals. Do not distribute these notes 170 Prof. R.T. M’Closkey, UCLA
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Discrete-time Sinusoids A discrete-time sinusoid is generated when a continuous-time sinusoid is sampled. For example, given frequency ω (which we will assume to be positive without loss of generality) the continuous-time (complex) sinusoid is defined as u ( t ) = e jωt , t ( -∞ , ) . If this signal is sampled with a sample period t s , the following discrete-time 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 continuous-time 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
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