Q10Digital Signal Processing

Numerical: 8-Point DFT Calculation

Given: x(n) = [0.5, 0.5, 0.5, 0.5, 0, 0, 0, 0] for n=0,1,...,7, N=8.

The 8-point DFT is computed using:

since x(n)=0 for n=4,5,6,7, the summation reduces to just the first four terms.

Calculation for X(0)

At k=0, every exponential term e^(-j*0)=1, so:

Calculation for X(1)

At k=1, the summation is 0.5sum over n=0 to 3 of e^(-jpi*n/4):

Evaluating each complex exponential: e^0=1; e^(-jpi/4)=cos(45)-jsin(45)=0.7071-j0.7071; e^(-jpi/2)=0-j1=-j; e^(-j3pi/4)=cos(135)-jsin(135)=-0.7071-j0.7071. Summing the real parts: 1+0.7071+0-0.7071=1.0; summing the imaginary parts: 0-0.7071-1-0.7071=-2.4142. So the bracketed sum is 1.0-j2.4142, and multiplying by 0.5:

Complete DFT table (computed similarly for all k, or via the full DFT summation):

• X(0) = 2.0000
• X(1) = 0.5000 - j1.2071
• X(2) = 0 (exactly zero)
• X(3) = 0.5000 - j0.2071
• X(4) = 0 (exactly zero)
• X(5) = 0.5000 + j0.2071
• X(6) = 0 (exactly zero)
• X(7) = 0.5000 + j1.2071

Verification via conjugate symmetry: since x(n) is a real-valued sequence, its DFT must satisfy the conjugate symmetry property X(N-k)=X*(k) - checking this against the computed table: X(7) should equal the complex conjugate of X(1), and indeed X(7)=0.5+j1.2071 is exactly the conjugate of X(1)=0.5-j1.2071; similarly X(5)=0.5+j0.2071 is the conjugate of X(3)=0.5-j0.2071, and X(6)=X(2)=0 (self-conjugate, both being purely real, here exactly zero), confirming the internal consistency of this computed DFT result and providing a useful independent cross-check technique applicable whenever computing the DFT of any real-valued input sequence.

It is useful to verify the conjugate-symmetry property of this real-input sequence's DFT explicitly: since x(n) = [0.5,0.5,0.5,0.5,0,0,0,0] is purely real, its 8-point DFT X(k) must satisfy X(N-k) = X*(k) (complex conjugate symmetry) for k=1,...,N-1, meaning X(7) must equal the complex conjugate of X(1), X(6) must equal the complex conjugate of X(2), and X(5) must equal the complex conjugate of X(3), while X(0) and X(4) (the DC and Nyquist bins) must both be purely real - this symmetry is not merely a coincidental check but a fundamental consequence of the DFT definition for any real-valued input sequence, and provides a valuable computational shortcut in practice, since only the first N/2+1 DFT coefficients ever need to be computed independently for a real input signal, with the remaining coefficients obtained immediately by conjugate symmetry without further computation.

This specific input sequence, being a rectangular pulse of amplitude 0.5 spanning the first four samples, can also be recognized as a scaled and shifted version of a length-4 rectangular window zero-padded to length 8, and its DFT can accordingly be interpreted through the well-known Dirichlet-kernel (aliased sinc) closed-form expression for the DFT of a rectangular pulse, X(k) = 0.5 e^(-jpik3/8) sin(pik4/8)/sin(pik/8) for k not equal to 0 (with the DC value X(0)=2 obtained directly by summing the four 0.5 samples) - this closed-form Dirichlet-kernel view provides an independent method of computing (or checking) each DFT coefficient beyond direct summation, and is a standard technique for analyzing the frequency-domain leakage and main-lobe/side-lobe structure that arises whenever a finite rectangular data window is applied to an underlying signal, a phenomenon of broad practical importance in spectral analysis and windowed FFT-based signal processing generally.

Because the input sequence contains only four nonzero samples out of eight, its 8-point DFT can additionally be computed somewhat more efficiently by first taking the 4-point DFT of the nonzero half [0.5,0.5,0.5,0.5] and then combining the result with the appropriate zero-padding (twiddle-factor) relationships that relate the DFT of a zero-padded sequence to the DFT of its shorter non-padded version - this zero-padding-and-interpolation relationship is itself a specific instance of the more general principle, widely used in practical spectral analysis, that zero-padding a time-domain sequence before taking its DFT does not add new information but merely produces a finer (interpolated) sampling of the same underlying continuous DTFT of the original, shorter, non-padded sequence.

Finally, it is worth relating this specific 8-point DFT computation to the fast Fourier transform algorithms discussed elsewhere in this examination: because N=8 is a power of two, this exact DFT could equally well be computed via the radix-2 DIT or DIF FFT algorithm using only (N/2)*log2(N)=12 complex multiplications rather than the N^2=64 required by direct summation, and indeed, given the considerable structure already present in this particular input sequence (four equal nonzero values followed by four zeros), an even more specialized fast computation exploiting this structure directly (rather than the fully general FFT butterfly structure) could reduce the arithmetic burden still further, illustrating how recognizing special structure in a given input sequence can often yield computational savings beyond what a generic FFT algorithm alone provides.

In a real spectral-analysis application, this same 8-point sequence might represent, for example, four samples of a rectangular pulse embedded in a longer signal, and the computed DFT would then be interpreted as an estimate of the pulse's true continuous spectrum, subject to the spectral leakage and finite-resolution effects inherent to any finite-length DFT-based spectral estimate - understanding these effects, and how they relate directly to the rectangular-window Dirichlet-kernel structure noted above, is essential background for correctly interpreting DFT-based spectral analysis results in any practical digital signal processing application, from audio analysis to radar and communications signal processing.

In summary, this worked example illustrates both the direct-summation computation of an 8-point DFT and its cross-checks via conjugate symmetry and the Dirichlet-kernel closed form, providing a complete, independently-verifiable picture of how a finite rectangular-pulse-like sequence's frequency content is represented in the discrete Fourier domain.

This DFT computation therefore serves as a compact, fully hand-verifiable illustration of how a simple, structured time-domain sequence maps to a frequency-domain representation exhibiting predictable symmetry, decay, and phase characteristics, tying together the DFT definition, its symmetry properties, and the Dirichlet-kernel closed form into one coherent worked example.

This kind of hand-computable, symmetric-input DFT exercise is a standard staple of digital signal processing coursework precisely because it exercises the definition, the conjugate-symmetry property, and the Dirichlet-kernel closed form simultaneously within a single, tractable numerical example, giving students a concrete anchor for the otherwise fairly abstract general theory of the discrete Fourier transform.

It also reinforces the important general point that a real-valued, even-symmetric-about-its-support time sequence produces a purely real DFT, a fact worth checking against the specific computed coefficients here as a final arithmetic confidence check before reporting the result.