Negative (basic concepts of signals and systems) and bilateral frequency spectrum

• Transfer from Beijing University of Posts and Telecommunications MOOC "Signals and Systems"
• For bilateral spectrum, negative frequency $n \ omega_1$ Only mathematical meaning no physical meaning
• Why introduce negative frequencies?
  $f(t)$ is a real function, there must be decomposed into an imaginary index Conjugate $jn ^ e {\} omega_1$ with $e ^ {- jn \ omega_1}$In order to ensure $f(t)$ real function invariant properties
  $c_ncos(n\omega_1t+\phi_n)=\frac{1}{2}c_n[e^{j(n\omega_1t+\phi_n)}+e^{-j(n\omega_1t+\phi_n)}]$
• Amplitude spectrum and a phase spectrum
  
• Practical examples
  
• 2 is divided into a magnitude spectrum, even symmetrical phase spectrum odd symmetry