1. Assume object does not vary in y
Signal equation for constant Gx gradient, no y gradient
Assume object does not vary in y
How does s(t) and F(m) relate?
For F(m), look first at F(0) and then F(1)? How does the phase across f(n) applied by the kernel change between these 2 Fourier points?
Between F(2) and F(1)?

2. Lecture 5 MR: Spatial Resolution, etc.
Readout vs. Phase Encode
Recall,
Readout Gx t t
Phase Encode
Gy Gy t ty
Each phase encode adds an incremental phase shift across the FOV.
Phase shift over FOVy

3. Phase encode number 0, 1, 2, 3, 4, 5, 6 y=FOV/2
Turn off the readout gradient and look at the phase of individual signal spins at 5 locations along the y axis. We will first look at the ky=0 phase encode and then move up in k-space. The phase of the spins at each y level are written for each phase encode below.
Phase encode number
0, 1, 2, 3, 4, 5, 6 5 4 3 2 1 Gy t
y=FOV/2
0, p, 2p, 3p, 4p, 5p, 6p
y=FOV/4
0, p/2, p, 3 p/2, 2p, 5p/2, 3p
y=0
0, 0, 0 , , , 0,
y=-FOV/4
0, -p/2, p,-3 p/2,-2p,-5p/2,-3p
y=-FOV/2
0, -p, -2p, -3p, -4p, -5p, 6p

4. Readout Direction Note: Each excitation line across kx (fast)
Next measure sample new ky (slow)
For a given time sampling rate, use an anti-aliasing filter (LPF)
I(t)
sreceive
LPF
A/D
cos(ot)
Bandwidth is set
so it’s width is

5. Readout Direction (2)
We should never have aliasing in readout direction.
FOV is free in the readout direction.
Note: If we have hardware (receiver speed), we can increase the sampling rate, increase the low pass filter cutoff, to reduce t
- If we reduce kx, what happens to FOV?
- SNR effects?
- Where should we place the long axis?

6. Aliasing in Phase Encode Direction
Referenced to 2D FT
Referred to as “phase wrap”
Left) ky = 1/FOV
Right) sampling ky = 2/FOV

7. Phase encode number 0, 1, 2, 3, 4, 5, 6 y=FOV/2
Turn off the readout gradient and look at the phase of individual signal spins at 5 locations along the y axis. We will first look at the ky=0 phase encode and then move up in k-space.
Phase encode number
0, 1, 2, 3, 4, 5, 6 5 4 3 2 1 Gy t
y=FOV/2
0, p, 2p, 3p, 4p, 5p, 6p
y=FOV/4
0, p/2, p, 3 p/2, 2p, 5p/2, 3p
y=0
0, 0, 0 , , , 0,
y=-FOV/4
0, -p/2, p,-3 p/2,-2p,-5p/2,-3p
y=-FOV/2
0, -p, -2p, -3p, -4p, -5p, 6p

8. Aliasing in Phase Encode Direction
Ignore x encoding and look at phase of individual spins after each
phase encoding step when ky = 2/FOV
Phase encode number
0, , , , 4, ,
y=FOV2
0, 2 p, 4 p, p , 8 p , 10 p, 12 p
y=FOV/4
0, p, 2p, p, 4p, p, 6p
y=0
0, 0, , , , ,
y=-FOV/4
0, -p , -2p, -3 p, -4p, -5p, -6p
y=-FOV/2
0, -2p, -4p, -6p, -8p, -10p, -12p

9. Why didn’t neck wrap around?
Sagittal Phase Wrap
Phase direction
Frequency Direction
Why didn’t neck wrap around?

10. Spatial Resolution Why? Main Lobe Width = in x, in y
Definition: Spatial Resolution Element
True for large kx
# of samples · spatial resolution = FOV

11. k-Space Acquisition ky kx Phase Direction One line of k-space
Encode
Sampled
Signal
DAQ kx ky
Phase
Direction
One line of k-space
acquired per TR
Frequency Direction

12. Fast Fourier Transform
FFT

13. Characteristics of Rectilinear Sampling
Unaliased FOV kx
Spatial Resolution ky

14. 512 x 512
8 x 8

15. 512 x 512
16 x 16

16. 512 x 512
32 x 32

17. 512 x 512
64 x 64

18. 512 x 512
128 x 128

19. 512 x 512
256 x 256

20. Resolution is lower, bigger pixels
Avoiding Aliasing
Phase Encode Direction
need “small”, but
1) If we increase Npe to compensate for reducing ky
scan time increases
Resolution depends on
FOV increases
2) Same Npe reducing ky reduce wky
What happens?
Resolution is lower, bigger pixels
FOV increases
same scan time

21. 2D FT or Spin Warp Gx t Tread Gy max Gy t Why this 2? extent in kx
about kx = 0
Gy max Gy t
Why this 2?

22. Fundamental Limits – What are they?
Limits to Resolution
- Max gradient strength (<4 G/cm)
(<1 G/cm)
Readout duration
As Tread , T2 decay blurs
FOVy vs.
For fixed scan time, improved resolution costs in FOV.
- Sequence-dependent effects
Gibbs Ringing
Fundamental Limits – What are they?