This is the first video in an 18-video series on Fourier Analysis of Discrete-Time Signals. In this first video we describe one of our primary goals, namely, writing a discrete-time signal as a weighted combination of complex exponentials. Trying to write a discrete-time signal in this form will eventually leads to the derivation of the Discrete-Time Fourier Series (DTFS) and Discrete-Time Fourier Transform (DTFT).

The DTFS is used to represent periodic discrete-time signals in the frequency domain. It's continuous-time counterpart studied previously is the Fourier Series (FS).

The DTFT is used to represent non-periodic discrete-time signals in the frequency domain. It's continuous-time counterpart studied previously is the Fourier Transform (FT).

The Discrete-Time Fourier Series (DTFS) can be used to write N0-periodic discrete-time signals x[k] as a weighted combination of complex exponentials. In this video, we reason through the form of the DTFS, namely:

1) The DTFS must consist of exponentials whose frequencies are some multiple of the fundamental frequency of the signal. If this was not the case, then the DTFS would not be periodic with the same period as the signal x[k].

2) If x[k] is N0-periodic, only N0 terms need to be included in the weighted combination. Since discrete-time complex exponentials are non-unique, including more than N0 terms would just be adding in additional exponentials that had already been included in the summation.

At the end of this video we now know the form of the DTFS equation. In the next video, we'll derive an equation that lets us to compute the DTFS coefficients (i.e. the weights in the summation).

In this video, we derive an equation for the Discrete-Time Fourier Series (DTFS) coefficients of the periodic discrete-time signal x[k]. Given this N0-periodic signal, the equation we derive lets us compute the N0 DTFS coefficients as a function of x[k]. In subsequent videos, we will use this equation to compute the DTFS coefficients for specific periodic discrete-time signals

This is the first of several examples of computing the Discrete-Time Fourier Series (DTFS). In this example, we find the DTFS of a sinusoid using the "inspection" technique. This approach doesn't use the equation for the DTFS coefficients, but instead uses trigonometric identities to directly manipulate the signal into a weighted combination of complex exponential signals. Once written in this form, the DTFS coefficients can just be "picked off" of the resulting expression. In the next video, we work the same example but use the DTFS equation directly.

Similar to the previous video, this video also works an example where we find the Discrete-Time Fourier Series (DTFS) of a sinusoid. However, in this example we evaluate the DTFS coefficient equation directly which requires us to simplify a more complicated summation and make use of a summation result that we previously derived.

Similar to the previous video, this video also works an example where we find the Discrete-Time Fourier Series (DTFS) of a sinusoid. However, in this example we evaluate the DTFS coefficient equation directly which requires us to simplify a more complicated summation and make use of a summation result that we previously derived.

In this last example we compute the DTFS coefficients of a periodic square wave. The square wave is parameterized by its width 2M+1, and it repeats every N0 samples. We directly evaluate the DTFS coefficient equation and then perform some algebraic simplifications to find a "nice" final expression for the Dr. In this case, it turns out that we can write the Dr as a ratio of sinusoids. We also plot the amplitude and phase spectrum of the signal for different values of M.

The previous videos in this series have examined the Discrete-Time Fourier Series (DTFS) which can be used to represent periodic discrete-time signals in the frequency domain. We use this frequency-domain representation of periodic signals as a starting point to derive the frequency-domain representation of non-periodic signals. This representation is called the Discrete-Time Fourier Transform (DTFT).

Given the non-periodic signal x[k], the DTFT is X(Omega). Having derived an equation for X(Omega), we work several examples of computing the DTFT in subsequent videos.

This and the next few videos work various examples of finding the Discrete-Time Fourier Transform of a discrete-time signal x[k]. In this video, we being with the simplest possible signal, namely, a signal that zero everywhere except for a single value at time k = 0 (e.g. x[k] is the unit impulse function delta[k]).

​We compute the DTFT of x[k] to yield X(Omega). We see that X(Omega) is constant for all frequencies. This establishes that a single impulse in the time domain is a constant in the frequency domain.

In the next few videos we continue working examples of the DTFT for increasingly more complicated signals.