Dear ALL !
A doubt was raised in the class yesterday as to why k.n must have a range 0 to N-1.
Here is the answer : While k indicates the discrete sample in the time domain and n indicates the sample in the frequency domain...both k and n vary between 0 and N-1. So k.n need not be between 0 and N-1.
But, when we calculate the Euler function using various values of k and n, you will see that all converge into the same set which is equivalent to taking kn between 0 and N-1.
You can try this by calculating the function for (k,n) = (0,0), (0,1), (0,2), (0,3), (1,0), (1,1), (1,2),(1,3),(2,0),(2,1),(2,2),(2,3), (3,0),(3,1),(3,2),(3,3).
All values will finally belong to the set kn = {0,1,2,3}.
Please go through the complete explanation by carefully noting the points.
Note : z-Transform, DFT and FFT each have its own advantages and disadvantages which we shall understand indepth once the mathematics is clear.
I will post again in the evening with details of a new quiz.
Good Day !
Hi Sir,
ReplyDeleteI have one doubt in my mind for a long time in N-point DFT related to the above explanation.
Assume the following parameters:
N-Number of Frequency Samples.
M-Number of Time samples
If length of the time samples is 'M', we should take the N -Point DFT to get the Frequency Spectrum and the condition is N>=M.
Questions:
1.Why is this condition N>=M?
2.What happen if we take N- DFT such that NM
4. Basically what is the relation between the number of samples in time domain and the number of frequency samples (for the given time samples )
Dinesh,
ReplyDelete1. As all real world signals are time limited, we will begin understanding DFT by sampling the signal x(t).
2. We know that when the sampling frequency is greater than the Nyquist rate, we can get back the original signal without aliasing effects.
3. When we select the sampling rate of fs (reciprocal of Ts : time between successive samples), we will get a frequency spectrum that is 'REPETITIVE' in nature.
4. The spectrum looks like a periodic signal (x axis being frequency )with a repetition rate of fs.
5. From ONE CYCLE of this spectrum we only need to select HALF period (ie from 0 to fs/2) as the remaining half (fs/2 to fs) is the mirror image of the first half.
6. So, We must select an N such that the frequency spectrum has zero magnitude around fs/2
7. The number of frequency samples will be LESS THAN or EQUAL TO the number of time samples.
8. The choice of number of samples depends on the practical considerations such as aliasing , computation time, accuracy required etc.
As I already mentioned in the class, the focus was mainly on the mathematics of DFT rather than on the concept of DFT...which I will explain through a practical example during our next meeting.
Keep posting
Good Day.
Thank you Sir. Sorry I missed the last class... We Will discuss more during the next meeting
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