DTFT Computation and Comparison with MATLAB Built-in Functions

The computation of the Discrete-Time Fourier Transform (DTFT) represents a fundamental component in digital signal processing. Understanding the derivation methodology of DTFT is essential for effective signal analysis and processing applications. To facilitate this understanding, I have developed the following DTFT computation source code that implements the transform using a direct summation approach across specified frequency points. This implementation calculates the DTFT through vectorized operations, where the input sequence is multiplied by complex exponentials at discrete frequency points. The code features parameterized frequency resolution control and supports both magnitude and phase spectrum analysis. Users can compare these custom DTFT results with MATLAB's built-in functions (such as freqz for rational transfer functions or fft for discrete frequency analysis) to validate implementation accuracy. The comparative analysis enables deeper insights into DTFT's theoretical foundations and practical applications in DSP. Key implementation aspects include handling finite-duration sequences, frequency vector normalization, and computational efficiency optimization through matrix operations rather than nested loops. The code also demonstrates proper handling of complex-valued results and visualization techniques for frequency domain representations. Through this comparison, users can verify numerical precision, understand windowing effects on spectral leakage, and analyze time-frequency resolution trade-offs inherent in Fourier transform implementations.