how to sample sine wave to 4 points dft output

Question

sinusoid x(t)=1+sin(2*pi *500) sampling rate :1000 samples/s Can I have 4 points DFT output with the above data.

Answer 1

$ x(t)=1+sin(2*pi *500)$ is a sinusoidal signal with frequency 500 Hz and the sampling rate is 1000 Hz. So you have 1000 samples/sec and 500 cycles/sec.

So you have only 2 samples/cycle. So you can calculate 4 point DFT by taking only 4 values (consecutive) from the samples (Since it represent 2 cycles).

For distortionless DFT you should take atleast one full cycle. Here 2 point DFT also possible. As the number of samples increases the DFT graph become more smooth.