Properties of DTFT I

Consider two discrete-time signals, each with their respective Discrete-Time Fourier Transforms DTFTs.

The signals are first multiplied by constants a and b, then combined to form a resultant signal.

Applying the DTFT to the resultant signal transforms it into a new DTFT, demonstrating the linearity property.

When a signal is delayed by a certain number of units, its DTFT experiences a phase shift proportional to the delay, known as the time-shifting property.

The frequency-shifting property occurs when a time-domain signal is multiplied by a complex exponential that shifts the signal’s frequency components.

Time reversal shows that reversing a discrete-time signal in time results in a frequency domain representation reflected about the vertical axis.

The conjugation property reveals that taking the complex conjugate of a signal results in both reflection and conjugation of its frequency components in the frequency domain.

When a signal is scaled by a factor k, it retains values only at intervals that are multiples of k.

Computing the DTFT of this signal compresses the frequency components by k, demonstrating the time scaling property.