1. Linearity property

Statement: If $x_1(n) \leftarrow FT \rightarrow X_1(\omega) and \ \ x_2(n) \leftarrow FT \rightarrow X_2(\omega)$

then, $a_1x_1(n)+a_2x_2(n) \leftarrow FT \rightarrow a_1 X_1(\omega) + a_2 X_2(\omega)$

Fourier transform of a linear combination of two or more signals is equal to the same linear combination of fourier transform of individual signal.

2. Periodicity

Statement: The periodicity is defined as $X(\omega) = X(\omega+2\omega k).$

If $x(n) \leftarrow FT \rightarrow X(k)$

Then, $1. x(n) = x(n+N), \\ 2. X(k) = X(k+N)$

3.Time shift

Statement: If $x(n) \leftarrow FT \rightarrow X(\omega) then, x(n-k) \leftarrow FT \rightarrow e^{-jωk}X(\omega)$

If a signal is shifted in time domain by k samples then the magnitude spectrum is unchanged but the phase spectrum is unchanged by amount $(-\omega k).$

4. Convolution

Statement: If $x_1(n) \leftarrow FT \rightarrow X_1(\omega) and x_2(n) \leftarrow FT \rightarrow X_2(\omega)$

then, $x_1(n) *x_2(n) \leftarrow FT \rightarrow X_1(\omega) * X_2(\omega)$

Convolution of two signals in time domain is equivalent to multiplication in frequency domain.

5. Time Reversal

Statement: If $x(n) \leftarrow FT \rightarrow X(\omega) then, x(-n) \leftarrow FT \rightarrow X(-\omega)$

If we fold the sequence in time domain then the magnitude spectrum is unchanged but the polarity of phase spectrum is unchanged.