--- title: "Time-scaling property of DTFS" subject: "Signals and Systems" chapter: 2 key: "semester_4/signals_and_systems:dtfs-time-scaling" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/dtfs-time-scaling site: BCT Engineering --- # Time-scaling property of DTFS Signals and Systems · Chapter 2: Fourier Series - Asked in: 2079 Asoj, 4 marks; 2079 Chaitra, 4 marks (asked 2 times) - Video: [L46 Signals and Systems Time Scaling and Time Shifting Properties of Fourier Series](https://www.youtube.com/watch?v=6H6_nMbckSU) — SAI RADHA KRISHNA GUJARATHI - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/dtfs-time-scaling - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-2 ## Question State and prove the time-scaling property for discrete-time Fourier series pair. ## Short answer Time expansion by integer m inserts m-1 zeros between samples, yielding x_{(m)}[n] with period mN. Its Fourier coefficients become b_k = a_k / m (with a_k repeating every N). Prove by summing only non-zero samples. ## Answer In discrete time, $x[an]$ is not defined for non-integer $an$, so time scaling is defined through the **time-expanded** signal (inserting zeros). #### Statement Let $x[n]$ be periodic with period $N$ and DTFS coefficients $a_k$. For a positive integer $m$ define $$ x_{(m)}[n] = \begin{cases} x[n/m], & n \text{ a multiple of } m \\ 0, & \text{otherwise} \end{cases} $$ ($m - 1$ zeros are inserted between samples). Then $x_{(m)}[n]$ is periodic with period $mN$ and $$ x_{(m)}[n] \overset{FS}{\longleftrightarrow} \frac{1}{m}a_k $$ (with $a_k$ viewed as periodic with period $N$, so the $mN$ coefficients are $a_k/m$ for $k = 0, 1, \dots, mN-1$). #### Proof The period of $x_{(m)}[n]$ is $mN$, so its coefficients are $$ b_k = \frac{1}{mN}\sum_{n=0}^{mN-1}x_{(m)}[n]e^{-jk\frac{2\pi}{mN}n} $$ Only $n = mr$ ($r = 0, 1, \dots, N-1$) give non-zero terms, where $x_{(m)}[mr] = x[r]$: $$ \begin{aligned} b_k &= \frac{1}{mN}\sum_{r=0}^{N-1}x[r]\,e^{-jk\frac{2\pi}{mN}mr} \\ &= \frac{1}{m}\left[\frac{1}{N}\sum_{r=0}^{N-1}x[r]e^{-jk\frac{2\pi}{N}r}\right] \\ &= \frac{1}{m}a_k \end{aligned} $$ #### Interpretation - Expanding in time by $m$ multiplies the period by $m$, so the fundamental frequency becomes $\omega_0/m = 2\pi/(mN)$ and the spectrum is packed $m$ times more densely. - The coefficient values are divided by $m$, because the same $N$ non-zero samples of one period are now averaged over $mN$ samples instead of $N$. - The coefficient pattern of $x[n]$ is repeated $m$ times within one period of $b_k$. **Example:** $x[n] = \{1, 1\}$ with $N = 2$ has $a_0 = 1$, $a_1 = 0$. With $m = 2$, $x_{(2)}[n] = \{1, 0, 1, 0\}$ ($N = 4$) gives $b_k = \frac{1}{4}(1 + e^{-j\pi k}) = \{0.5, 0, 0.5, 0\}$, which is $\frac{1}{2}\{a_0, a_1, a_0, a_1\}$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is x_(m)[n] in DT time scaling? A. x[mn] B. x[n/m] when n is a multiple of m and 0 otherwise C. x[n+m] D. m x[n] Answer: B. Zeros are inserted between samples; m-1 zeros between each pair. (Key point: Time expansion) 2. What is the period of x_(m)[n]? A. N/m B. N C. mN D. N+m Answer: C. Each of the N samples is spread over m positions, giving period mN. (Key point: New period) 3. What are the coefficients of x_(m)[n]? A. m a_k B. a_k C. a_k / m D. a_(mk) Answer: C. Same sum over N non-zero samples divided by mN instead of N gives (1/m) a_k. (Key point: Coefficient result) 4. Which samples contribute in the proof? A. Only n = mr, where x_(m)[mr] = x[r] B. All n from 0 to mN-1 C. Only n = 0 D. Only odd n Answer: A. The inserted zeros contribute nothing, leaving N terms. (Key point: Proof key step) ## Close variants set in other exams - [Time-shifting property of DTFS](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-time-shifting) — Signals and Systems, chapter 2, asked 5× - [Time-scaling property of CTFS](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-time-scaling) — Signals and Systems, chapter 2, asked 4× ## Related in Signals and Systems - [Derive the discrete-time Fourier series](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-derivation) — Signals and Systems, chapter 2, asked 15× - [Frequency shifting property of DTFS](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-frequency-shifting) — Signals and Systems, chapter 2, asked 3× - [Convolution property of DTFS](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-convolution-property) — Signals and Systems, chapter 2 - [CTFS time shifting and time scaling](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-shift-scale-properties) — Signals and Systems, chapter 2, asked 2× - [FS coefficients; time scaling and conjugation](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-significance-scaling-conjugation) — Signals and Systems, chapter 2 - [DTFS linearity and time shift](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-linearity-time-shift) — Signals and Systems, chapter 2 ## Related in other subjects - [Fourier transform property: Scaling](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-scaling) — Communication Systems, chapter 1 - [Fourier transform property: Time shifting](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-time-shifting) — Communication Systems, chapter 1 - [Properties of continuous-time Fourier series](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afourier-series-properties) — Data Communication, chapter 3 - [Autocorrelation and ESD as Fourier transform pair](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aautocorrelation-esd-fourier-pair) — Communication Systems, chapter 1