Time domain <==> Frequency domain

Example #1(a): simple sinusoid

Continuous-time waveform 
[Graphics:../Images/index_gr_97.gif]

[Graphics:../Images/index_gr_98.gif]

[Graphics:../Images/index_gr_99.gif]
Sampled continuous-time waveform 
[Graphics:../Images/index_gr_100.gif]
[Graphics:../Images/index_gr_101.gif]

[Graphics:../Images/index_gr_102.gif]

Discrete spectrum: Fast Fourier Transform (FFT) 
[Graphics:../Images/index_gr_103.gif]

[Graphics:../Images/index_gr_104.gif]

[Graphics:../Images/index_gr_105.gif]

[Graphics:../Images/index_gr_106.gif]

Example #1(b): simple sinusoid with DC offset

Continuous-time waveform 
[Graphics:../Images/index_gr_107.gif]

[Graphics:../Images/index_gr_108.gif]

[Graphics:../Images/index_gr_109.gif]
Sampled continuous-time waveform 
[Graphics:../Images/index_gr_110.gif]
[Graphics:../Images/index_gr_111.gif]

[Graphics:../Images/index_gr_112.gif]

Discrete spectrum: Fast Fourier Transform (FFT) 
[Graphics:../Images/index_gr_113.gif]

[Graphics:../Images/index_gr_114.gif]

[Graphics:../Images/index_gr_115.gif]

[Graphics:../Images/index_gr_116.gif]

Example #2: sum of two sinusoids

Continuous-time waveform 
[Graphics:../Images/index_gr_117.gif]

[Graphics:../Images/index_gr_118.gif]

[Graphics:../Images/index_gr_119.gif]
Sampled continuous-time waveform 
[Graphics:../Images/index_gr_120.gif]
[Graphics:../Images/index_gr_121.gif]

[Graphics:../Images/index_gr_122.gif]

Discrete spectrum: Fast Fourier Transform (FFT) 
[Graphics:../Images/index_gr_123.gif]

[Graphics:../Images/index_gr_124.gif]

[Graphics:../Images/index_gr_125.gif]

[Graphics:../Images/index_gr_126.gif]

Example #3: product of two sinusoids

Continuous-time waveform 
[Graphics:../Images/index_gr_127.gif]

[Graphics:../Images/index_gr_128.gif]

[Graphics:../Images/index_gr_129.gif]
Sampled continuous-time waveform 
[Graphics:../Images/index_gr_130.gif]
[Graphics:../Images/index_gr_131.gif]

[Graphics:../Images/index_gr_132.gif]

Discrete spectrum: Fast Fourier Transform (FFT) 
[Graphics:../Images/index_gr_133.gif]

[Graphics:../Images/index_gr_134.gif]

[Graphics:../Images/index_gr_135.gif]

[Graphics:../Images/index_gr_136.gif]

Does this make sense? 
[Graphics:../Images/index_gr_137.gif]
[Graphics:../Images/index_gr_138.gif]

Note that the expression 'sin(2π t) sin(60π t)' can also be expressed as '1/2 [(cos(58π t)-cos(62π t)]'. Thus two frequencies are present in the signal 29Hz and 31Hz.

Example #4: square wave

Continuous-time waveform 
[Graphics:../Images/index_gr_139.gif]

[Graphics:../Images/index_gr_140.gif]

Sampled continuous-time waveform 
[Graphics:../Images/index_gr_141.gif]
[Graphics:../Images/index_gr_142.gif]

[Graphics:../Images/index_gr_143.gif]

Discrete spectrum: Fast Fourier Transform (FFT) 
[Graphics:../Images/index_gr_144.gif]

[Graphics:../Images/index_gr_145.gif]

Example #5: single pulse

Continuous-time waveform 
[Graphics:../Images/index_gr_146.gif]

[Graphics:../Images/index_gr_147.gif]

Sampled continuous-time waveform 
[Graphics:../Images/index_gr_148.gif]
[Graphics:../Images/index_gr_149.gif]

[Graphics:../Images/index_gr_150.gif]

Discrete spectrum: Fast Fourier Transform (FFT) 
[Graphics:../Images/index_gr_151.gif]

[Graphics:../Images/index_gr_152.gif]

[Graphics:../Images/index_gr_153.gif]

[Graphics:../Images/index_gr_154.gif]

Converted by Mathematica      January 13, 2003