In this module, we will derive an expansion for arbitrary discrete-time functions, and in doing so, derive the Discrete Time Fourier Transform (DTFT).

Since complex exponentials are eigenfunctions of linear time-invariant (LTI) systems, calculating the output of an LTI system ℋℋ given ej⁢ω⁢n ω n as an input amounts to simple multiplication, where ω 0 =2⁢π⁢kN ω 0 2 k N , and where H⁢k∈C H k is the eigenvalue corresponding to k. As shown in the figure, a simple exponential input would yield the output

Figure 1: Simple LTI system.

Using this and the fact that ℋℋ is linear, calculating y⁢n y n for combinations of complex exponentials is also straightforward.

c 1 ⁢ej⁢ ω 1 ⁢n+ c 2 ⁢ej⁢ ω 2 ⁢n→ c 1 ⁢H⁢ k 1 ⁢ej⁢ ω 1 ⁢n+ c 2 ⁢H⁢ k 2 ⁢ej⁢ ω 1 ⁢n c 1 ω 1 n c 2 ω 2 n c 1 H k 1 ω 1 n c 2 H k 2 ω 1 n ∑l c l ⁢ej⁢ ω l ⁢n→∑l c l ⁢H⁢ k l ⁢ej⁢ ω l ⁢n l c l ω l n l c l H k l ω l n

The action of HH on an input such as those in the two equations above is easy to explain. ℋℋ independently scales each exponential component ej⁢ ω l ⁢n ω l n by a different complex number H⁢ k l ∈C H k l . As such, if we can write a function y⁢n y n as a combination of complex exponentials it allows us to easily calculate the output of a system.

Now, we will look to use the power of complex exponentials to see how we may represent arbitrary signals in terms of a set of simpler functions by superposition of a number of complex exponentials. Below we will present the Discrete-Time Fourier Transform (DTFT). Because the DTFT deals with nonperiodic signals, we must find a way to include all real frequencies in the general equations. For the DTFT we simply utilize summation over all real numbers rather than summation over integers in order to express the aperiodic signals.

It can be demonstrated that an arbitrary Discrete Time-periodic function f⁢n f n can be written as a linear combination of harmonic complex sinusoids

The c n c n - called the Fourier coefficients - tell us "how much" of the sinusoid ej⁢ ω 0 ⁢k⁢n j ω 0 k n is in f⁢n f n . The formula shows f⁢n f n as a sum of complex exponentials, each of which is easily processed by an LTI system (since it is an eigenfunction of every LTI system). Mathematically, it tells us that the set of complex exponentials ej⁢ ω 0 ⁢k⁢n  ,   k∈Z    k k j ω 0 k n form a basis for the space of N-periodic discrete time functions.

Because complex exponentials are eigenfunctions of LTI systems, it is often useful to represent signals using a set of complex exponentials as a basis. The discrete time Fourier transform synthesis formula expresses a discrete time, aperiodic function as the infinite sum of continuous frequency complex exponentials.