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| Keywords: Aliasing between a positive and a negative frequency.png en This figure depicts two complex sinusoids colored gold and cyan that fit the same sets of real and imaginary sample points They are thus aliases of each other when sampled at the rate f<sub>s</sub> indicated by the grid lines The gold-colored function depicts a positive frequency because its real part the cos function leads its imaginary part by 1/4 of one cycle The cyan function depicts a negative frequency because its real part lags the imaginary part 2013-03-27 01 09 45 own Bob K Octave script <source lang Matlab > graphics_toolkit gnuplot gold 251 159 3/256; arbitrary color choice sam_per_sec 1; T 1/sam_per_sec; sample interval dt T/20; time-resolution of continuous functions cycle_per_sec sam_per_sec/8; sam_per_sec 8 cycle_per_sec satisfies Nyquist figure subplot 3 1 1 xlim -2 10 ylim -1 3 1 3 Plot cosine function start_time_sec -2; stop_time_sec 10; x start_time_sec dt stop_time_sec; y cos 2 pi cycle_per_sec x ; plot x y 'color' gold 'linewidth' 4 box off no border around plot please hold on same axes for next 3 plots Plot sine function start_time_sec 0; stop_time_sec 10; x start_time_sec dt stop_time_sec; y sin 2 pi cycle_per_sec x ; plot x y 'color' gold 'linewidth' 4 Sample cosine function at sample-rate 1/T start_time_sec 0; stop_time_sec 8; x start_time_sec T stop_time_sec; y cos 2 pi cycle_per_sec x ; plot x y 'color' 'black' ' ' Sample sine function y sin 2 pi cycle_per_sec x ; plot x y 'color' 'black' ' ' set gca xaxislocation zero set gca yaxislocation zero set gca 'xgrid' 'on' ; set gca 'ygrid' 'off' ; set gca 'ytick' 0 ; set gca 'xtick' 0 8 ; subplot 3 1 2 xlim -2 10 ylim -1 3 1 3 cycle_per_sec2 cycle_per_sec - sam_per_sec; negative frequency Re-plot same cosine function on new axes start_time_sec -2; stop_time_sec 10; x start_time_sec dt stop_time_sec; y cos 2 pi cycle_per_sec x ; plot x y 'color' gold 'linewidth' 4 box off hold on Plot other cosine function start_time_sec -2; stop_time_sec 10; x start_time_sec dt stop_time_sec; y cos 2 pi cycle_per_sec2 x ; plot x y 'color' 'cyan' 'linewidth' 4 Sample cosine functions at sample-rate 1/T start_time_sec 0; stop_time_sec 8; x start_time_sec T stop_time_sec; y cos 2 pi cycle_per_sec x ; plot x y 'color' 'black' ' ' set gca xaxislocation zero set gca yaxislocation zero set gca 'xgrid' 'on' ; set gca 'ygrid' 'off' ; set gca 'ytick' 0 ; set gca 'xtick' 0 8 ; subplot 3 1 3 xlim -2 10 ylim -1 3 1 3 Re-plot original sine function on new axes start_time_sec 0; stop_time_sec 10; x start_time_sec dt stop_time_sec; y sin 2 pi cycle_per_sec x ; plot x y 'color' gold 'linewidth' 4 box off hold on Plot other sine function start_time_sec -2; stop_time_sec 10; x start_time_sec dt stop_time_sec; y sin 2 pi cycle_per_sec2 x ; plot x y 'color' 'cyan' 'linewidth' 4 Sample sine functions at sample-rate 1/T start_time_sec 0; stop_time_sec 8; x start_time_sec T stop_time_sec; y sin 2 pi cycle_per_sec x ; plot x y 'color' 'black' ' ' set gca xaxislocation zero set gca yaxislocation zero set gca 'xgrid' 'on' ; set gca 'ygrid' 'off' ; set gca 'ytick' 0 ; set gca 'xtick' 0 8 ; </source> cc-zero Uploaded with UploadWizard Aliasing Negative frequency Waveforms | ||||
