Discrete-Time Fourıer Transform, Lecture notes of Fourier Transform and Series

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THE DISCRETE- TIME FOURIER

TRANSFORM

• (^) Continuous Fourier Transform

• (^) Euler`s identity:
• (^) Trigonometrical equation for DFT

• (^) X(m)- m-th DFT output component , i.e. X(0),…
• (^) m-the index of DFT output in frequency domain
• (^) m=0,1,…, N-1.
• (^) x[n] – the sequence of input samples x[0], x[1],…
• (^) n-the time domain index of input samples n=0,1,2, …, N-1.
• (^) j=
• (^) N- number of samples of the input sequences and number of frequency points in the DFT output

• (^) If we are sampling a continuous signal at a rate of 500 samples/second and, then, perform a 16-point DFT on the sampled data, the fundamental frequency of the sinusoids is or 31.25.

• (^) The other X(m) analysis frequencies are integral multiples of the fundamental frequency :
• (^) x 31.25 =0 Hz
• (^) x 31.25 =31.25 Hz
• (^) x 31.25 =62.5 Hz
• (^) x 31.25 =93.75 Hz

• (^) Magnitude function (magnitude spectrum) is determined as

Example

• (^) Let’s say we want to sample and perform an 8-point DFT on a continuous input signal containing components at 1 kHz and 2 kHz, expressed as

• (^) With a sample rate of , we sample the input every 1/ = seconds. Because N = 8, we need 8 input sample values on which to perform the DFT.