One of the greatest things about Digital Signal Processing is that if you have a computer, and you probably do have one since your watching this video lectures, if you have a computer, you have a fully functional signal processing lab at your fingertips. And more importantly, a computer also has interfaces that will allow you to visualize the signals that you create by writing a few lines of code. You can visualize them as plots on the screen, and more importantly, and that's really what got me into signal processing, you can materialize the signals as audio signals that you can hear with your ears. Now we need an interface because our ears are analog devices, whereas the PC is a digital device. And we will see later in this class the precise details of this transformation from digital to analog. Here, let's just go through a hand waving overview to give you the fundamentals of digital to analog conversion. So that, in future modules we will be able to use audio examples that you can listen to. So let's start with a simple digital sinusoid. You pick your favorite trigonometric functions, say a sin, digital frequency Omega zero, measured in radians and an initial phase if you want, again measured in radians. And your discrete-time sequence is then computed very easily with a loop. At each step, you increment index n and you compute the sin of omega 0 n + theta. And you will get a series of samples that looks like this. How would a sinusoid like this sound? Well, before we can answer this we have to bridge the gap between the digital representation and the analog representation that we present to our ears. Now remember, in discrete time n has no physical dimension. And the periodicity of a wave form is determined by how many samples you have to wait before the pattern repeats. In the physical world on the other hand, the periodicity is measured in how many seconds you have to wait before the pattern repeats and the frequency is measured in hertz which is a physical unit of one over seconds. The PC bridges this gap via a sound card, a device that takes a series of samples, and builds an electric signal that we can feed to a loud speaker. And at the heart of the sound card is a system clock, with a period TS measured in seconds. This TS is a time that we wait before we take a new sample from a discrete time sequence, and feed it in the sound card. So, if TS is the time in seconds between samples, a periodicity of capital M samples in the digital domain will become a periodicity of MTS seconds in the physical domain, so that the real world frequency of a periodic pattern like a sinusoid in the digital domain will become 1 over MTS hertz in the physical domain. So let's give an example. Usually we choose FS, the number of samples per second that we use in the sound card rather than TS, but the relationship is very simple. TS is equal to one over FS and of course FS is equal to one over TS. A typical value for FS is 48 kilohertz, this is probably familiar to you because it's the sampling frequency of say a DVD, so that TS is about 20 micro seconds. Now assume we built a sinusoid in the digital domain with the periodicity of 110 samples. If we fed the sinusoid to a sound card at 48 kilohertz, we would get sinusoid with a perceivable pitch of 440 hertz, which is the standard concert pitch in Western music.