Satellite Access Phase 2 Update
Satellite Access Phase 2 progress and objectives for normative phase completion. Discussion on mobility management, power saving, and system behavior in 5G satellite access. Importance of collaborative mindset and upcoming meetings for finalizing specifications.
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Presentation Transcript
Continuous-Time Fourier Transform
Content Introduction Fourier Integral Fourier Transform Properties of Fourier Transform Convolution Parseval s Theorem
Continuous-Time Fourier Transform Introduction
The Topic Continuous Time Discrete Time Periodic Discrete Fourier Transform Fourier Series Continuous Fourier Transform Aperiodic Fourier Transform
Review of Fourier Series Deal with continuous-time periodic signals. Discrete frequency spectra. A Periodic Signal f(t) t T 2T 3T
Two Forms for Fourier Series Sinusoidal Form 2 2 a 2 nt nt = = = + + 0 ( ) cos sin f t a b n n T T 1 1 n n 2 / 2 T = ( ) cos a f t n tdt 0 n T 2 2 / 2 T / 2 T T = ( dt ) a f t 0 T / 2 / 2 T = ( ) sin b f t n tdt 0 n T / 2 T 1 Complex Form: = n / 2 T = jn t = jn t ( ) f t c ne ( ) c f t e dt 0 0 n T / 2 T
How to Deal with Aperiodic Signal? A Periodic Signal f(t) t T If T , what happens?
Continuous-Time Fourier Transform Fourier Integral
Fourier Integral 1 / 2 T = = jn t ( ) c f t e dt = jn t ( ) f t c e 0 0 n T T n T / 2 T n 2 1 T 1 / 2 T = n = =2 0 = jn jn t ( ) f e d e 0 0 0 T T T / 2 T 1 / 2 T = n = jn jn t ( ) f e d e 0 0 0 T 2 2 / 2 T = = Let 0 T 1 / 2 T = n = jn jn t ( ) f e d e 0 0 T 2 / 2 T = 0 T d 1 = j j t ( ) f e d e d T 2
Fourier Integral 1 = j j t ( ) ( ) f t f e d e d 2 F(j ) 1 d Synthesis = j t ( ) ( ) f t F j e 2 Analysis = j t ( ) ( ) F j f t e dt
Fourier Series vs. Fourier Integral Fourier Series: = n Period Function = jn t ( ) f t c ne 0 1 / 2 T Discrete Spectra = jn t ( ) c f t e dt 0 n T T / 2 T Fourier Integral: Non-Period Function 1 d = j t ( ) ( ) f t F j e 2 Continuous Spectra = j t ( ) ( ) F j f t e dt
Continuous-Time Fourier Transform Fourier Transform
Fourier Transform Pair Inverse Fourier Transform: 1 d Synthesis = j t ( ) ( ) f t F j e 2 Fourier Transform: Analysis = j t ( ) ( ) F j f t e dt
Existence of the Fourier Transform Sufficient Condition: f(t) is absolutely integrable, i.e., | ( | ) f t dt
Continuous Spectra = j t ( ) ( ) F j f t e dt FI(j ) = + ( ) = ( ) j ( ) F j F j jF j R I ( ) ( ) je | ( | ) F FR(j ) Phase Magnitude
Example f(t) 1 t -1 1 1 1 1 = j t = ( ) ( ) = F j f t e dt j t j t e e 1 dt j 1 2 sin j = = j j ( ) e e
Example 3 2 F( ) 1 0 f(t) -1 -10 -5 0 5 10 1 3 2 t |F( )| -1 1 1 1 0 1 j -10 -5 0 5 10 1 = j t = ( ) ( ) = F j f t e dt j t j t e e 1 dt 4 arg[F( )] 1 2 -5 2 sin -10 j = = j j ( ) e e 0 0 5 10
Example f(t) e t t = j t ( ) ( ) = F j f t e dt t j t e e dt 0 1 = + = ( ) j t e dt + j 0
Example f(t) 1 =2 e t |F(j )| 0.5 t 0 -10 -5 0 5 10 2 = j t ( ) ( ) = F j f t e dt t j t e e dt arg[F(j )] 0 0 1 -2 = + = ( ) j t -10 -5 0 5 10 e dt + j 0
