C williams w alexander ncsu the discrete fourier

C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 139 / 193

The DFT for Spectral Estimation
We note several things
POINT 1:
Even with large L , the spectrum of a rectangular window have
very large sidelobes
This causes a distortion that cannot easily be eliminated by
increasing L
POINT 2:
Signal are often of some set duration L
In most practical applications, we cannot easily and arbitrarily
increase L
C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 140 / 193

The DFT for Spectral Estimation
To address these problem, other window functions, w ( n ) can be
used
These windows have slightly different spectral characteristics that
the spectrum of the rectangular window
An example of such a window is a Hanning window
w ( n ) =
(
1
2
( 1 - cos
2 π
L - 1
n ) , 0 ≤ n ≤ L - 1
0 , otherwise
(93)
The sidelobes of the Hanning window are significantly smaller that
those of the rectangular window
Its main lobe, however, is approximately twice as wide
C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 141 / 193

Resolving Close Frequencies
The size of the main lobe of the window function determines
whether or not two adjacent frequencies can be resolved
Consider a signal that is a sum of two sinusoids
x ( n ) = cos( ω
0
n ) + cos( ω
1
n ) (94)
For a rectangular window of length L , this results in the following
spectrum
ˆ
X ( ω ) = π [ W ( ω - ω
0
)+ W ( ω - ω
1
)+ W ( ω + ω
0
)+ W ( ω + ω
1
)] (95)
C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 142 / 193

Resolving Close Frequencies
Distortion can cause us not to be able to distinguish one spectral
line from the other
This occurs if | ω
0
- ω
1
| <
2 π
L
If | ω
0
- ω
1
| <
2 π
L
, the two window functions W ( ω - ω
0
) and
W ( ω - ω
1
) overlap
As a result, the two spectral lines of x ( n ) become indistinguishable
If | ω
0
- ω
1
| ≥
2 π
L
, we will see the two distinct peaks corresponding
to the individual spectral lines
NOTE: This is the case for a rectangular window. This becomes
worse if the main lobe is wider, as is the case for the Hanning
window
C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 143 / 193

Resolving Close Frequencies
Example (7.8)
Consider the following signal
x ( n ) = cos( ω
0
n ) + cos( ω
1
n ) + cos( ω
2
n ) (96)
where ω
0
= 0 . 2 π , ω
1
= 0 . 22 π , and ω
2
= 0 . 6 π
Let N = 2
15
. Plot the DFT for L = 25 , 50 , and 100
For the sampling frequency F
s
= 1 kHz , these frequencies
correspond to F
0
= 100 Hz , F
1
= 110 Hz , and F
2
= 300 Hz
C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 144 / 193

Resolving Close Frequencies
Example (7.8)
The figure below show the resulting spectrum for N = 2
15
, L = 25
0 50 100 150 200 250 300 350 400 450 500
0
5
10
15
20
25
F (Hz)
|X(k)|
Spectrum of Finite Duration Signal of x(n), N = 2
15
, L = 25
Figure: Spectrum of Finite Duration of x ( n ) .
C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 145 / 193

Resolving Close Frequencies
Example (7.8)
The figure below show the resulting spectrum for N = 2
15
, L = 50
0 50 100 150 200 250 300 350 400 450 500
0
5
10
15
20
25
30
35
F (Hz)
|X(k)|
Spectrum of Finite Duration Signal of x(n), N = 2
15
, L = 50
Figure: Spectrum of Finite Duration of x ( n ) .