#05 Signals & Systems #05 | DT Scaling & Reversal: Decimation, Upsampling & Time Flip

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

1. 1. Time reversal

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

   y[n]=x[−n] reflects each sample's index about zero.

   The value at zero remains at zero; a sample originally at k appears at −k.

   Narration transcript

   In the previous lesson, we learned how to shift discrete-time signals. Now we introduce two more operations: time reversal and time scaling. Time reversal of x of n gives us x of negative n. To draw x of negative n, we flip the signal around the origin, n equals zero. Each sample at position n moves to position negative n. The value at n equals zero stays in place, acting as the pivot point. This operation is also called folding or reflection.

2. 2. Reversal example

   Original values [1,−0.5,2,0.5,−1,1] occupy indices −2 through 3.

   Reversal gives [1,−1,0.5,2,−0.5,1] at indices −3 through 2, with value 2 still at zero.

   Narration transcript

   Let us see reversal in action. Consider x of n with values one, negative zero point five, two, zero point five, negative one, and one, defined from n equals negative two to n equals three. To find x of negative n, we reverse the index: the value at n equals negative two goes to n equals two. The value at n equals negative one goes to n equals one. And so on. The value at n equals zero stays at zero. After reversal, the signal reads: one, negative one, zero point five, two, negative zero point five, one, now defined from n equals negative three to n equals two.

3. 3. Integer decimation

   For positive integer M, y[n]=x[Mn] selects original indices divisible by M and relabels them by division by M.

   Every output index n is integer; evaluate the original sequence at Mn.

   Narration transcript

   Time scaling for discrete-time signals works differently from continuous-time. For y of n equals x of M n, we multiply each index by M, then keep only the values that fall on integer positions. The systematic method is: for each value of n in the output, compute M times n. If the result is an integer index of the original signal, that value survives. Otherwise, it is discarded. This process is called decimation or downsampling when M is greater than one.

4. 4. Correct the worked decimation

   Original values [−1,0.5,1,−1,0.5,2,3,−1,2] occupy indices −3 through 5.

   For x[2n], keep original indices −2,0,2,4.

   The result is [0.5,−1,2,−1] at output indices −1,0,1,2. There is no fifth retained sample at output index 3.

   Narration transcript

   Let us find x of two n using the divide-and-check method. Given x of n with values negative one, zero point five, one, negative one, zero point five, two, three, negative one, two, defined from n equals negative three to n equals five. For each index in x of n, we check: can n be divided by two to give an integer? n equals negative three: negative three over two is not an integer, so cross it out. n equals negative two: negative two over two equals negative one, which is an integer, so keep it. We continue this process. The surviving values form x of two n: zero point five, negative one, two, negative one, two, defined from n equals negative one to n equals three.

5. 5. Zero-insertion upsampling

   Define y[n]=x[n/2] at even n, and define y[n]=0 at odd n.

   The odd-index rule is an explicit extension, since the original sequence has no noninteger samples. Upsampling alone does not reconstruct missing information.

   Narration transcript

   What about x of n over two? This is called upsampling. When n is even, n over two is an integer, so we use the value from x of n over two. When n is odd, n over two is not an integer, so we insert zero. The result stretches the signal by a factor of two, with zeros between the original samples. This creates a longer sequence with the original values preserved at even indices.

6. 6. Repeat the index check

   Form a table of original index k, divisibility by M, output index k/M and retained value.

   The exact number retained depends on support and index alignment; roughly half remain for factor two on a long contiguous support.

   Narration transcript

   Here is another decimation example to reinforce the method. Given a different signal x of n, we want to find x of two n. We apply the same divide-and-check procedure: go through each index, divide by two, and keep only those that produce integer results. The key insight is that decimation always reduces the number of samples. If the original signal has N samples, x of two n will have roughly N over two samples. Information between the kept samples is permanently lost.

7. 7. Review

   

   Reversal negates indices; integer decimation selects divisible original indices; zero-insertion upsampling supplies an explicit value at new positions.

   Keep the origin and support visible in every result.

   Narration transcript

   Let us summarize what we covered. Time reversal x of negative n flips the signal around n equals zero. Time scaling x of M n uses the divide-and-check method: divide each index by M, keep only integer results. Decimation x of two n discards every other sample. Upsampling x of n over two inserts zeros between samples. In the next lesson, we will combine these operations and explore periodic discrete-time signals.

Source video: Signals & Systems #05 | DT Scaling & Reversal: Decimation, Upsampling & Time Flip (4:28)