#04 Signals & Systems #04 | Discrete-Time Signals: Definition, Sampling & Shifting

Below are all the lines written in the notebook together with the full narration transcript.

1. 1. Discrete-time signals

   Corrected solution notes. The audio is the original recording and may contain errors or incomplete phrases. Where they differ, use the corrected written solution.

   A sequence x[n] is defined at integer indices. Values between indices are not part of the sequence unless an interpolation rule is separately specified.

   Use isolated stems to display its samples.

   Narration transcript

   In our first lecture, we worked with continuous-time signals, where the signal is defined for every real value of t. Now we move to a fundamentally different type: discrete-time signals. A discrete-time signal x of n is defined only at integer values of n. Between these integers, the signal simply does not exist. We represent discrete-time signals using stem plots, where each value appears as a dot at the top of a vertical line. This is very different from the connected curves we drew for continuous-time signals.

2. 2. Example sequence

   At indices −3,−2,−1,0,1,2,3, the values are 0.5,−2,0.5,1,2,1.5,0.5.

   In an ordered vector, the fourth entry corresponds to index zero.

   Narration transcript

   Let us look at a concrete example. Consider x of n with the following values: at n equals negative three, the value is zero point five. At n equals negative two, it is negative two. At n equals negative one, zero point five. At n equals zero, one. At n equals one, two. At n equals two, one point five. And at n equals three, zero point five. Each value is an isolated point. We can also write this compactly using vector notation: x of n equals the sequence zero point five, negative two, zero point five, one, two, one point five, zero point five, with an arrow marking the position of n equals zero.

3. 3. Vector convention

   List entries from left to right in increasing index order, and mark the entry at n=0.

   The pointer convention matters when the vector entries are held fixed while their indices change.

   Narration transcript

   Vector notation is a convenient shorthand for discrete-time signals. We list the values in order and place an arrow underneath to mark which element corresponds to n equals zero. Values to the left of the arrow correspond to negative n, and values to the right correspond to positive n. This compact representation will be especially useful when we perform operations like shifting and scaling, because we can track how the n equals zero position moves.

4. 4. Sampling

   x[n]=xc(nTs), with Ts>0 and integer n.

   Sampling records values at selected times. A smaller interval increases sampling density; exact reconstruction still requires appropriate bandwidth and sampling assumptions.

   Narration transcript

   How do we obtain discrete-time signals in practice? Through sampling. Given a continuous-time signal x sub c of t, we sample it at regular intervals of T sub s seconds. The result is x of n equals x sub c of n times T sub s, where n is an integer. Graphically, imagine placing vertical stems at every T sub s seconds along the continuous curve, and reading off the values at those points. The continuous curve between samples is discarded. The sampling period T sub s determines how finely we capture the original signal. Smaller T sub s means more samples and better representation.

5. 5. Delay: correct pointer direction

   For y[n]=x[n−1], each plotted sample moves one index right.

   In a fixed left-to-right value vector, the zero-index pointer moves one entry left.

   Example: values [−1,2,0.5,3,0.5] initially at indices −2 through 2 become indexed −1 through 3; y[0]=2.

   Narration transcript

   Now let us learn our first discrete-time operation: shifting. Given x of n, we can create a delayed version y of n equals x of n minus n zero, where n zero is a positive integer. This shifts the entire signal to the right by n zero positions. In vector notation, this is equivalent to shifting the n equals zero pointer to the right. For example, given x of n equals the sequence negative one, two, zero point five, three, zero point five with n equals zero at the third element. To find x of n minus one, we shift the n equals zero pointer one position to the right. The signal values stay the same, but they now occur one step later.

6. 6. Advance

   For y[n]=x[n+1], plotted values move one index left, while the zero pointer in a fixed value vector moves one entry right.

   In the same example y[0]=x[1]=3. Substitute an index to check the rule.

   Narration transcript

   We can also advance a signal by shifting it to the left. For y of n equals x of n plus n zero, we shift the n equals zero pointer to the right by n zero positions, which effectively moves the signal to the left. For x of n plus one, the pointer shifts right by one. For x of n plus two, it shifts right by two. In general: to draw x of n plus n zero, shift the n equals zero pointer to the right by n zero. To draw x of n minus n zero, shift the pointer to the left by n zero. This convention is the same as what we learned for continuous-time signals, but now we only deal with integer shifts.

7. 7. Review

   

   Delay moves the plotted sequence right but its zero pointer left in a fixed ordered vector; advance reverses those directions.

   Always state whether values or the index marker are being moved.

   Narration transcript

   Let us review what we have covered. Discrete-time signals are defined only at integer values of n, and we represent them using stem plots or vector notation with an n equals zero marker. We obtain them from continuous-time signals through sampling, where x of n equals x sub c of n times T sub s. And we can shift them in time: x of n minus n zero delays the signal by n zero steps, while x of n plus n zero advances it. In the next lesson, we will learn two more operations: time scaling and time reversal for discrete-time signals.

Source video: Signals & Systems #04 | Discrete-Time Signals: Definition, Sampling & Shifting (5:00)