#09 Signals & Systems #09 | Signal Relationships - Derivative & Integral Chains

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

1. 1. Distributional derivative chain

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

   For r(t)=t u(t), differentiation gives u(t) almost everywhere and in the distributional sense.

   Differentiating the unit jump gives δ(t); hence the second distributional derivative of the ramp is δ(t).

   Narration transcript

   Now we connect the three elementary signals through calculus. Starting with the ramp r of t, its derivative is the unit step: d r of t d t equals u of t. The ramp rises linearly, so its slope is one for positive time and zero for negative time, which is the step. Taking one more derivative: d u of t d t equals delta of t. The step has a sudden jump at the origin, and the derivative of a jump is an impulse. Combining both: delta of t equals d squared r of t d t squared. The impulse is the second derivative of the ramp.

2. 2. Integration chain

   Accumulating a unit impulse gives a unit step, with the isolated endpoint convention stated separately.

   Integrating the step from negative infinity gives the ramp. These integrals converge for the causal elementary signals here.

   Narration transcript

   Now in reverse. Starting with delta of t and integrating: the integral of delta of tau d tau from negative infinity to t gives u of t. The impulse has zero area until we cross t equals zero, then the accumulated area becomes one. Integrating once more: the integral of u of tau d tau from negative infinity to t gives r of t. We accumulate the constant one, producing a linearly increasing function. So: delta integrates to step, step integrates to ramp, and in reverse, ramp differentiates to step, step differentiates to delta.

3. 3. Correct discrete identities

   With u[0]=1 and r[n]=n u[n], the backward difference is r[n]−r[n−1]=u[n−1].

   The inclusive running sum of u[k] is (n+1)u[n]; the sum through n−1 is r[n].

   The forward difference r[n+1]−r[n] is u[n].

   Narration transcript

   In discrete time, differences replace derivatives and sums replace integrals. The first backward difference of the unit step gives the Kronecker delta: delta of n equals u of n minus u of n minus one. At n equals zero: u of zero minus u of negative one equals one minus zero equals one. At all other n, the values cancel. For the ramp: r of n equals the cumulative sum of u of k, giving zero, one, two, three. The first difference of r of n gives back u of n.

4. 4. Impulse from the step

   $\displaystyle \delta \left[n\right]=u\left[n\right]-u\left[n-1\right].$

   At zero the difference is one; everywhere else it is zero. This is a Kronecker impulse with a finite sample value.

   Narration transcript

   Let us verify visually. Draw u of n as a stem plot: zeros for negative n, ones from n equals zero onward. Now draw u of n minus one: ones start at n equals one. Subtracting: at n equals zero, one minus zero equals one. At n equals one onward, one minus one equals zero. The result is delta of n: a single stem at n equals zero with value one. This is the discrete-time equivalent of delta of t equals d u of t d t.

5. 5. Compare operations

   Continuous-time derivatives and integrals correspond to discrete differences and sums, but index offsets must be retained.

   A CT Dirac impulse is a distribution of unit area; it is not an ordinary point value to be substituted as infinity.

   Narration transcript

   Let us organize into a comparison table. Unit step: CT has u of t line plot, DT has u of n stem plot. Impulse: CT delta of t is the arrow with unit area, DT delta of n is value one at origin. Ramp: CT r of t equals t u of t line, DT r of n equals n u of n stems. The derivative chain in CT becomes a difference chain in DT, and integration becomes summation.

6. 6. Six elementary signals

   Draw CT step and ramp as curves, and DT versions as isolated stems.

   Mark a CT impulse by its weight and location; a DT impulse is a unit sample. Their normalization rules differ.

   Narration transcript

   Here are all six elementary signals together: CT and DT versions of step, impulse, and ramp. Continuous-time signals use curves and lines, discrete-time signals use isolated stems. The impulse is unique: infinite in CT, finite in DT. These six functions are the building blocks for all of signal processing.

7. 7. Review

   

   CT ramp, step and impulse form a distributional derivative chain.

   In DT, use the explicit forward or backward difference and exact summation limit; omitting the one-sample shift changes the result.

   Narration transcript

   This concludes Lecture Three. We defined the unit step for switching, the ramp for linear growth, and the impulse with its sifting, multiplication, and scaling properties. We connected them through derivatives and integrals: ramp to step to impulse, and integration reverses the chain. These relationships will be essential for system responses and convolution in upcoming lectures.

Source video: Signals & Systems #09 | Signal Relationships - Derivative & Integral Chains (4:08)