--- title: "4-point DFT of {2, 1+j, 1−j}" subject: "Signals and Systems" chapter: 3 key: "semester_4/signals_and_systems:dft-complex-sequence" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/dft-complex-sequence site: BCT Engineering --- # 4-point DFT of {2, 1+j, 1−j} Signals and Systems · Chapter 3: Fourier Transform - Asked in: 2070 Magh, 4+2 marks - Video: [Computation of 4 point DFT (Example 2)](https://www.youtube.com/watch?v=LzVXaVivwAY) — EnggClasses - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/dft-complex-sequence - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3 ## Question Compute 4-point DFT of a signal x[n] = {2, 1+j, 1−j} and plot its magnitude and phase spectrums. ## Short answer Zero-pad to length 4: x[n]={2, 1+j, 1−j, 0}. Using W4=−j, X[0]=4, X[1]=2, X[2]=2−2j, X[3]=2j. Magnitudes: 4, 2, 2.83, 2; phases: 0°, 0°, −45°, 90°. ## Answer The sequence has only 3 samples, so append one zero for a 4-point DFT: $x[n] = \{2,\; 1+j,\; 1-j,\; 0\}$. **Formula:** $X[k] = \sum_{n=0}^{3}x[n]W_4^{nk}$, with $W_4 = e^{-j\pi/2} = -j$, so $W_4^0=1$, $W_4^1=-j$, $W_4^2=-1$, $W_4^3=j$. $$ \begin{bmatrix} X[0] \\ X[1] \\ X[2] \\ X[3] \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & -j & -1 & j \\ 1 & -1 & 1 & -1 \\ 1 & j & -1 & -j \end{bmatrix} \begin{bmatrix} 2 \\ 1+j \\ 1-j \\ 0 \end{bmatrix} $$ **Working:** $$ \begin{aligned} X[0] &= 2 + (1+j) + (1-j) + 0 = 4 \\ X[1] &= 2 + (1+j)(-j) + (1-j)(-1) + 0 = 2 + (1 - j) + (-1 + j) = 2 \\ X[2] &= 2 - (1+j) + (1-j) - 0 = 2 - 2j \\ X[3] &= 2 + (1+j)(j) + (1-j)(-1) + 0 = 2 + (-1 + j) + (-1 + j) = 2j \end{aligned} $$ **Answer:** $X[k] = \{4,\; 2,\; 2-2j,\; 2j\}$ | $k$ | $X[k]$ | $\lvert X[k]\rvert$ | $\angle X[k]$ | |---|---|---|---| | 0 | $4$ | 4 | $0^\circ$ | | 1 | $2$ | 2 | $0^\circ$ | | 2 | $2-2j$ | $2\sqrt2 = 2.83$ | $-45^\circ$ | | 3 | $2j$ | 2 | $90^\circ$ | **Spectra (stem plots):** ```text |X[k]| 4 | | 3 | | | 2 | | | | | 1 | | | | | 0 +--+---+---+---+--> k 0 1 2 3 phase (degrees) 90 | | 45 | | 0 +--o---o---+---+--> k -45 | | 0 1 2 3 ``` (Here $x[n]$ is complex, so $X[3] \ne X^*[1]$; the spectrum has no conjugate symmetry.) **Check (Parseval):** $\sum|x[n]|^2 = 4 + 2 + 2 + 0 = 8$ and $\frac{1}{4}\sum|X[k]|^2 = \frac{1}{4}(16 + 4 + 8 + 4) = 8$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is a 3-sample sequence handled in a 4-point DFT? A. Repeat the last sample B. Discard one sample C. Use a 3-point DFT instead D. Append a zero Answer: D. The sequence becomes {2, 1+j, 1-j, 0}. (Key point: Zero padding) 2. What is X[0]? A. 4 B. 2 C. 2+2j D. 0 Answer: A. 2 + (1+j) + (1-j) = 4. (Key point: X[0]) 3. What is X[2]? A. 2 + 2j B. 2 - 2j C. 2 D. -2j Answer: B. X[2] = 2 - (1+j) + (1-j) = 2 - 2j. (Key point: X[2]) 4. Why is X[3] not the conjugate of X[1]? A. The sequence is too short B. The DFT is wrong C. x[n] is complex, so conjugate symmetry does not apply D. N is even Answer: C. Only real sequences give X[N-k] = X*[k]. 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