Presented By: Abhishek Verma DTFT Discrete Time Fourier Transform (DTFT) is Fourier transform of discrete time sequence represented by complex exponential sequence π βπππ where π is the real frequency variable. It maps time domain sequence into a continuous and periodic function of frequency variable. βπππ π(π ππ ) = β π₯(π)π π=ββ This equation represents DTFT. It can be applied on any arbitrary sequence. π₯(π) = This equation represents IDTFT. 1 π ππ )π πππ ππ π(π 2π βπ DFT Discrete Fourier Transform (DFT) is a finite duration discrete frequency sequence which is obtained by sampling one period of Fourier transform. Sampling is done at N equally spaced points, over the period extending from π = 0 π‘π π = 2π. βπ2πππ/π π(π) = πβ1 π=0 π₯(π)π For k=0,1,2β¦.N-1 This equation represents N point DFT. 1 πβ1 π₯(π) = π(π)π π2πππ/π π=0 π For n=0,1,2β¦.N-1 This equation represents N point IDFT. Twiddle Factor ππ = π βπ2π/π This is called twiddle factor. It makes computation of DFT a bit easy and fast. ππ π(π) = πβ1 π₯ π π π π=0 For k=0,1,2β¦.N-1 This equation represents N point DFT. 1 πβ1 βππ π₯(π) = π(π)π π π π=0 For n=0,1,2β¦.N-1 This equation represents N point IDFT. DFT Properties 1. Periodicity π₯ π =π₯ π+π π π = π[π + π] 2. Linearity π1 π₯1 π + π2 π₯2 π π1 π1 π + π2 π2 π 3. Time Reversal π₯[π β π] π[π β π] 4. Circular time shift π₯[π β π]π X[π]π βπ2πππ/π DFT Properties 5. Circular frequency shift π₯[π]π π2πππ/π π[π β π]π 6. Circular convolution π₯1 [π] β π₯2 [π] π1 π π2 [π] 7. Circular correlation π₯ π β π¦ β βπ π π π β [π] 8. Multiplication π₯1 [π]π₯2 [π] 1 π1 [π] β π2 [π] π DFT Properties 9. Complex conjugate π₯ β [π] π β [π β π] 10. Parsevalβs Theorem πβ1 π₯ π π¦β π π=0 πβ1 1 π π π πβ π π=0 Computational Complexity in DFT In direct computation, π(π) = πβ1 π=0 π₯ π ππ For k=0,1,2β¦.N-1 We have Complex multiplications π β π = π2 Complex additions π π β 1 = π2 β π ππ FFT Fast Fourier Transform is used to reduce number of arithmetic operations involved in the computation of DFT by using: 1. Radix-2 Decimation in Time(DIT) algorithm 2. Radix-2 Decimation in Frequency(DIF) algorithm It uses following properties of twiddle factor: 1. Periodic ππ π+π = ππ π 2. Symmetric π π+ 2 ππ = βππ π Radix-2 DIT algorithm Radix-2 DIF algorithm Computational Complexity in FFT ππ+1 [π] = ππ [π] + ππ π ππ [π] π ππ+1 [π] = ππ π β ππ ππ [π] We have Complex multiplications π log 2 π 2 Complex additions π log 2 π Resolution in FFT When using FFT analysis to study the frequency spectrum of signals, there are limits on resolution between different frequencies, and on detectability of a small signal in the presence of a large one. There are two basic problems: 1. The fact that we can only measure the signal for a limited time 2. The fact that the FFT only calculates results for certain discrete frequency values (the 'FFT bins'). The sampled time waveform input to an FFT determines the computed spectrum. If an arbitrary signal is sampled at a rate equal to ππ over an acquisition time T, N samples are acquired. Then Resolution Frequency is given by: 1 π = Maximum Resolvable Frequency ππππ₯ ππππ₯ ππ π ππ = 2 Minimum Resolvable Frequency ππππ ππππ = 0 The following strategies achieve a finer frequency resolution: 1. Decrease the sampling frequency, fs. Decreasing fs usually is not practical because decreasing fs reduces the frequency range. 2. Increase the number of samples, N. Increasing N yields an increased number of lines over the original frequency range. Analyze a signal that contains two tones at 1,000 Hz and 1,100 Hz, use a sampling frequency of 10,000 Hz. Acquire data for 10 ms with a frequency resolution of 100 Hz. The following front panel shows the results of this analysis. Increase the acquisition time to 1 s to achieve a frequency resolution of 1 Hz. The following front panel shows the results obtained with a 1 s acquisition time. Rectangular Window The rectangular window takes all samples with equal weight into account. The main lobe of its magnitude spectrum is narrow, but the level of the side lobes is rather high. It has the highest frequency selectivity. ππ π = 1 ,0 < π < π β 1 0 , πππ ππ€βπππ Triangular Window For an odd window length 2Nβ1 , the triangular window can be expressed as the convolution of two rectangular windows. The main lobe is wider as for the rectangular window, but the level of the side lobes decays faster. 2π ππ π = 1 β , π β€πβ1 πβ1 Hanning Window The Hanning window is a smooth window whose first and last value is zero. It features a fast decay of the side lobes. πβπ 1 2ππ π = 1 β cos 2 πβ1 Hamming Window The Hamming window is a smooth window function whose first and last value is not zero. The level of the side lobes is approximately constant. πβπ 2ππ π = 0.54 + 0.46 cos π β 1 0 , πππ ππ€βπππ Blackman Window The Blackman window features a rapid decay of side lobes at the cost of a wide main lobe and low frequency selectivity. 2ππ 4ππ ππ΅ π = 0.42 + 0.5 cos + 0.08 cos πβ1 πβ1 Applications 1. It can be used in spectral analysis. 2. It can be used in signal processing. 3. It can be used in image processing.