--- title: "DTFT properties; convolution property" subject: "Signals and Systems" chapter: 3 key: "semester_4/signals_and_systems:dtft-properties-convolution" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/dtft-properties-convolution site: BCT Engineering --- # DTFT properties; convolution property Signals and Systems · Chapter 3: Fourier Transform - Asked in: 2082 Kartik, 6 marks; 2082 Bhadra (new course), 3+3 marks (asked 2 times) - Video: [Properties of Discrete Time Fourier Transform | Convolution Property | Signals and Systems](https://www.youtube.com/watch?v=IBl-oUYg760) — Ekeeda - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/dtft-properties-convolution - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3 ## Question List out (explain) the properties of Discrete Time Fourier Transform. State and prove the convolution property of Discrete Time Fourier Transform. ## Short answer Convolution in time corresponds to multiplication in frequency: y[n] = x[n]*h[n] <-> Y(e^{jw}) = X(e^{jw})H(e^{jw}). Proof: Y(e^{jw}) = sum_n (sum_k x[k]h[n-k]) e^{-jw n} = sum_k x[k] e^{-jw k} H(e^{jw}) = X(e^{jw})H(e^{jw}). ## Answer #### Properties of the DTFT Let $x[n] \leftrightarrow X(e^{j\omega})$ and $y[n] \leftrightarrow Y(e^{j\omega})$. | Property | Time domain | Frequency domain | |---|---|---| | Periodicity | $x[n]$ | $X(e^{j(\omega+2\pi)}) = X(e^{j\omega})$ | | Linearity | $ax[n] + by[n]$ | $aX(e^{j\omega}) + bY(e^{j\omega})$ | | Time shift | $x[n - n_0]$ | $e^{-j\omega n_0}X(e^{j\omega})$ | | Frequency shift | $e^{j\omega_0 n}x[n]$ | $X(e^{j(\omega-\omega_0)})$ | | Conjugation | $x^*[n]$ | $X^*(e^{-j\omega})$ | | Time reversal | $x[-n]$ | $X(e^{-j\omega})$ | | Differencing | $x[n] - x[n-1]$ | $(1 - e^{-j\omega})X(e^{j\omega})$ | | Differentiation in freq. | $nx[n]$ | $j\frac{dX(e^{j\omega})}{d\omega}$ | | Convolution | $x[n]*y[n]$ | $X(e^{j\omega})Y(e^{j\omega})$ | | Multiplication | $x[n]y[n]$ | $\frac{1}{2\pi}\int_{2\pi}X(e^{j\theta})Y(e^{j(\omega-\theta)})d\theta$ | | Parseval | $\sum\lvert x[n]\rvert^2$ | $\frac{1}{2\pi}\int_{2\pi}\lvert X(e^{j\omega})\rvert^2d\omega$ | Also, for real $x[n]$ the DTFT is conjugate symmetric: $X(e^{-j\omega}) = X^*(e^{j\omega})$, so the magnitude is even and the phase is odd. #### Convolution property: statement $$ y[n] = x[n]*h[n] = \sum_{k=-\infty}^{\infty}x[k]h[n-k] \;\xleftrightarrow{DTFT}\; Y(e^{j\omega}) = X(e^{j\omega})H(e^{j\omega}) $$ #### Proof $$ Y(e^{j\omega}) = \sum_{n=-\infty}^{\infty}\sum_{k=-\infty}^{\infty}x[k]h[n-k]e^{-j\omega n} = \sum_{k}x[k]\sum_{n}h[n-k]e^{-j\omega n} $$ Put $m = n - k$ in the inner sum: $$ \sum_{m=-\infty}^{\infty}h[m]e^{-j\omega(m+k)} = e^{-j\omega k}H(e^{j\omega}) $$ So $$ Y(e^{j\omega}) = H(e^{j\omega})\sum_{k}x[k]e^{-j\omega k} = X(e^{j\omega})H(e^{j\omega}) $$ **Use:** an LTI system's output spectrum is its input spectrum multiplied by the frequency response $H(e^{j\omega})$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The DTFT of x[n - n0] is: A. e^{jωn0} X(e^{jω}) B. e^{-jωn0} X(e^{jω}) C. X(e^{j(ω-n0)}) D. n0 X(e^{jω}) Answer: B. A delay adds a linear phase -ωn0. (Key point: Time shift) 2. The DTFT of e^{jω0 n} x[n] is: A. X(e^{j(ω-ω0)}) B. X(e^{j(ω+ω0)}) C. e^{jω0} X(e^{jω}) D. X(e^{jω}) e^{-jω0 n} Answer: A. Multiplying by a complex exponential shifts the spectrum by ω0. (Key point: Frequency shift) 3. What does the substitution m = n - k give in the proof? A. Inner sum = e^{-jωk} H(e^{jω}) B. Inner sum = H(e^{jωk}) C. Inner sum = h[k] D. Inner sum = X(e^{jω}) Answer: A. Then Σ h[m]e^{-jω(m+k)} = e^{-jωk}H(e^{jω}), and summing against x[k] gives X H. (Key point: Convolution proof) 4. The output spectrum of a DT LTI system is: A. The input spectrum multiplied by H(e^{jω}) B. The input spectrum convolved with H(e^{jω}) C. The input spectrum plus H(e^{jω}) D. Independent of the input Answer: A. Y = X H, so the output spectrum is the input spectrum shaped by the frequency response. 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